ELECTRICITY  AND  MAGNETISM 


» 


BY 

ERIC   GERARD 

Director  of  Institut  Electrotechnique  Montefiore^ 
University  of  Liege,  Belgium 


TRANSLATED  FROM  THE  FOURTH  FRENCH  EDITION 

UNDER  THE  SUPERVISION  OF  DR.  LOUIS  DUNCAN, 

BY 

R.   C.    DUNCAN 


NEW  YORK 

McGRAW   PUBLISHING   COMPANY 

114  LIBERTY  STREET 
1897 


Engineering 
Library 


Copyright,  1897, 

BY 
THE  W.  J.  JOHNSTON  COMPANY. 


PUBLISHERS'  NOTE. 


IT  is  scarcely  five  years  since  the  original  work  of  Pro- 
fessor Gerard  appeared,  but  in  that  time  it  has  reached  the 
fourth  edition  in  the  French,  and  has  been  translated  into 
the  German.  The  work  has  evidently  become  a  classic  in 
Europe  ;  and  in  view  of  this  fact,  and  of  the  further  consider- 
ation that  it  occupies  a  place  by  itself  in  electrical  literature, 
the  publishers  have  deemed  it  wise  to  bring  out  an  Amer- 
ican edition. 

The  original  intention  of  the  author  was  to  produce  a 
work  which,  while  avoiding  on  the  one  hand  the  shortcom- 
ings of  the  more  elementary  works,  would  not,  on  the  other 
hand,  be  so  difficult  to  read  as  to  be  only  intelligible  to  the 
favored  few.  How  well  he  has  succeeded  in  this  intention 
may  best  be  judged  by  the  favorable  reception  of  the  book 
abroad. 

The  present  work  is  a  translation  of  the  fourth  French 
edition  by  Mr.  J.  P.  Duncan,  under  the  supervision  of  Dr. 
Louis  Duncan  of  the  Johns  Hopkins  University  and  pres- 
ident of  the  American  Institute  of  Electrical  Engineers.  All 
parts  of  the  work  relating  to  the  general  subject  of  elec- 
tricity have  been  retained,  but  the  chapters  on  special  sub- 
jects, such  as  storage  batteries,  transformers,  and  other 
electrical  machinery,  have  been  omitted  for  two  reasons  :  first, 
because  the  information  contained  in  them  is  easily  acces- 
sible in  other  well-known  works  (which  is  not  the  case  with 


257732 


IV  PUBLISHERS'   NOTE. 

the  parts  that  have  been  retained)  ;  and  second,  because  the 
descriptions  of  particular  machinery  and  apparatus  refer 
almost  exclusively  to  European  practice,  which  is  in  some 
cases  quite  different  from  American  practice.  Again,  the 
saving  of  space  made  by  these  omissions  allows  the  publish- 
ers to  add  much  valuable  new  matter.  There  is  a  chapter 
on  hysteresis  and  molecular  magnetic  friction  by  Mr.  Charles 
P.  Steinmetz,  a  well  known  authority  on  the  subject.  The 
short  section  of  the  original  work  on  units  and  dimensions 
is  replaced  by  a  chapter  written  by  Dr.  Gary  T.  Hutchin- 
son,  which  gives  a  comprehensive  view  of  the  theory.  A 
chapter  on  impedance  by  Dr.  A.  E.  Kennelly,  is  another 
valuable  addition  to  the  work,  as  it  makes  plain  those  points 
in  the  theory  of  alternating  currents  which  give  the  average 
student  the  most  trouble. 


CONTENTS. 


CHAPTER  I.     Introduction. 

PAGE   SECTION 

1  2     Fundamental  units. 

2  2     Derived  units. 

2  3     Example  of  a  derived  unit. 

3  4     Dimensions  of  a  derived  unit. 
3  5     Mechanical  derived  units. 

5  6     Principle  of  the  conservation  of  energy. 

7  7     Multiples  and  submultiples  of  the  units. 

7  8     Application  of  the  dimension  of  units. 

GENERAL   THEOREMS    RELATIVE  TO    CENTRAL   FORCES. 

8  Q     Definitions. 

9  10     Elementary  law  governing  the  Newtonian  forces. 

10  ii     Field  of  force. 

11  12     Potential. 

14         13     Equipotential  surfaces. 

14  14     Case  of  a  single  mass. 

15  15     Uniform  field. 

15  16  Case  of  two  acting  masses. 

17  17  Tubes  of  force. 

17  18  Flux  of  force. 

18  19  Theorem  of  flux  of  force  in  a  tube  of  infinitely  small  section. 

18  20     Gauss'  theorem. 

19  21     Corollary  I. 

20  22     Corollary  II. 

20  23     Corollary  III. 

21  24     Unit  tube.     Number  of  lines  of  force. 

21         25     Potential  energy  of  masses  subjected  to  Newtonian  forces. 

APPLICATIONS. 

23         26     I.  An  infinitely  thin  homogeneous  spherical  shell  exercises 
no  action  upon  a  mass  within  it. 

v 


VI  CONTENTS. 

PAGE   SECTION 

25  27     II.  The  action  of  a  homogeneous  spherical  shell  on  an  ex- 

ternal point  is  the  same  as  if  the  whole  mass  were  con° 
centrated  at  the  centre  of  the  sphere. 

26  28     Action  of  a  homogeneous  sphere  upon  an  external  point, 

27  29     Action  of  a  homogeneous  sphere  upon  an  internal  point, 
27         30     Surface  pressure. 

29         31     Potential  due  to  an  infinitely  thin  disk  uniformly  charged. 

CHAPTER  II.     Magnetism. 

31  32  Definitions. 

31  33  Action  of  the  earth  on  a  magnet. 

31  34  Law  of  magnetic  attractions. 

32  35  Unit  pole. 

33  36  Definitions. 

34  37  Action  of  a  uniform  field  on  a  magnet. 

35  38  Terrestrial  magnetic  field. 

36  39  Weber's  hypothesis. 

38         40     Elementary  magnets.     Intensity  of  magnetizatiou. 

40  41     Magnetic  or  solenoidal  filament. 

41  42     Uniform  magnets. 

43  43  Magnetic  shells. 

44  44  Corollary. 

45  j.t;  Energy  of  a  shell  in  a  field. 

46  46  Relative  energy  of  two  shells. 
46  47  Artificial  magnets. 

50        48     Determination    of    the    magnetic    moment    of   i   magnet 

Magnetometer. 
53         49     Remarks. 
53         50     Measurement  of  angles. 

INDUCED   MAGNETIZATION. 

56  51  Magnetic  and  diamagnetic  bodies. 

56  52  Coefficient  of  Magnetization  or  Magnetic  Susceptibility. 

57  53  Cases  of  a  sphere  and  a  disc. 
59  54  Case  of  a  ring. 

59  55     Case  of  a  cylinder  of  indefinite  extent. 

60  56     Portative  power  of  a  magnet. 

61  57     Variations  of  intensity  of  magnetization  with  the  magnet**- 

ing  force.     Hysteresis. 
66         58     Frolich's  formula. 

66  59     Formula  of  Miiller,  von  Waltenhofen,  and  Kapp. 

67  60     Another  way  of  looking  at  induced  magnetization,  magnetic 

induction,  and  permeability. 


CONTENTS.  Vll 

PAGE  SECTION 

71  5i  Work  spent  in 'magnetizing. 

74  62  Numerical  results. 

78  63  Effect  of  temperature  on.  magnetism.     Recalescence. 

80  64  Ewing's  addition  to  Weber's  hypothesis. 

84  65  Equilibrium  of  a  body  ii&a  magnetic  field. 

CHAPTER  III.     Hysteresis  and  Molecular  Magnetic  Friction. 

87  66  Hysteresis. 

93  67  Hysteretic  loop. 

95  68  Molecular  magnetic  friction. 

97  69  Determination  of  hysteresis  and  molecular  magnetic  fric- 
tion. 

101  70  Loss  of  energy. 

104  71  Coefficient  of  hysteresis. 

107  72  Eddy  currents. 

109  73  Effect  of  molecular  magnetic  friction  and  eddy  currents, 

no  74  Equivalent  sine  curves. 

115  75  Hysteretic  losses. 

CHAPTER  IV.     Electricity. 

117  76  Phenomenon  of  electrification. 

118  77  Conductors  and  insulators. 

119  78  Electrification  by  influence. 

120  79  Quadrant  electrometer. 

121  80  Experiments. 

123  81  Distribution  on  a  conductor. 

124  82  Law  of  electric  actions. 

125  83  Definitions.     Electric  field.     Electric  potential. 

125  84  Potential  of  a  conductor  in  equilibrium. 

126  85  Potential  of  the  earth. 

127  86  Coulomb's  theorem. 

128  87  Electrostatic  pressure. 

128  88  Corresponding  elements. 

129  89  Power  of  points. 

129  90  Electric  screen. 

130  91  Lightning-rods. 

CONDENSERS — DIELECTRICS. 

130  92  Capacity  of  conductors. 

131  93  Condensers.     Spherical  condenser. 

132  94  Plate  condenser. 

r33  95  Guard-ring  condenser. 

'34  96  Absolute  electrometer. 


Vlll  CONTENTS. 

PAGE   SECTION 

135  97     Cylindrical  condenser,  one  plate  connected  to  earth. 

136  98     Cylindrical  condenser  disconnected  from  earth. 

136  99     Leyden  jar. 

137  100  Energy  of  electrified  conductors. 

138  101  Theory  of  the  quadrant  electrometer, 

139  102  Quadrant  electrometer  formula?. 

140  103  Specific  inductive  capacity  of  dielectrics. 

141  104  Nature  of  the  coefficient  k  in  Coimmo's  law. 

142  105  Role  of  the  dielectric.     Displacements. 

145  106     Residual  charge  of  a  condenser. 

146  107     Electromotive  force  of  contact.     Distinction  between  elec- 

tromotive force  and  difference  of  potential. 

ELECTRIC    DISCHARGES   AND   CURRENTS. 

148  108     Convective  discharge. 

149  109     Conductive  discharge.     Electric  current. 

151       no    Disruptive   discharge.      Electric   spark    and   brush  ;    their 
effects. 

LAWS   OF   THE   ELECTRIC   CURRENT. 

154  in     General  considerations. 

155  112     Law  of  successive  contacts. 

155  113     Thermal   and   chemical    electromotive    forces.      Means   of 

keeping  up  a  constant  difference  of  potential  in  a  con- 
ductor. 

156  114     Ohm's  law. 

157  115     Case  of  a  conductor  of  constant  section. 

158  116     Graphic  representation  of  Ohm's  law. 

159  117     Variable  period  of  the  current. 

160  118     Application  of  Ohm's  law  to  the  variable  period  of  the  cur- 

rent in  but  slightly  conductive  bodies. 

161  119     Application  of  Ohm's  law  to  the  case  of  a  heterogeneous 

circuit. 

162  120     Graphic  representation. 

163  121     Kirchhoff  s  laws. 

165  122     Application  to  derived  circuits. 

166  123     Wheatstone's  bridge  or  parallelogram. 

ENERGY   OF   THE   ELECTRIC  CURRENT. 

167  124     General  expression. 

168  125     Application  to  the  case  of  a  homogeneous  conductor.    Joule's 

effect. 
168       126     Case  of  heterogeneous  conductors.     Peltier  effect. 


CONTENTS.  IX 

JAGH    SECTION 

169  127     Chemical  effect  of  the  current.     Faraday's  and  Becquerel's 

laws. 

170  128     Grothiiss'  hypothesis. 

171  129     Application  of  the  conservation  of  energy  to  electrolysis: 

Voltaic  cell, 

THERMO-ELECTRIC   COUPLES. 

174  130  Seebeck  and  Peltier  effects. 

176  131  Kelvin  effect. 

178  132  Laws  of  thermo-electric  action. 

178  133  Thermo-electric  powers. 

183  134  Thermo-electric  pile. 

CHAPTER  V.     Electromagnetism. 

MAGNETIC   PHENOMENA   DUE   TO    CURRENTS. 

184  135     Oersted's  discovery. 

185  136     Magnetic  field  due  to  an  indefinite  rectilinear  current. 

186  137     Laplace's  law. 

189  138     Action  of  a  magnetic  field  on  an  element  of  current. 

190  139     Work    due   to   the   displacement  of   an  element  of  current 

under  the  action  of  a  pole. 

191  140    Work  due  to  the  displacement  of  a  circuit  under  the  action 

of  a  pole. 

192  141     Magnetic  potential  due  to  a  circuit.     Unit  of  current.     Am- 

pere's hypothesis  on  the  nature  of  magnetism. 

196  142     Energy  of  a  current  in  a  magnetic  field.     Maxwell's  rule. 

197  143     Relative  energy  of  two  currents. 

197  144     Intrinsic  energy  of  a  current. 

198  145     Faraday's  rule. 

APPLICATIONS  RELATING  TO   THE   MAGNETIC   POTENTIAL   OF  THE  CURREN1/ 

200  146  Case  of  an  indefinite  rectilinear  current. 

202  147  Case  of  a  circular  current.     Tangent-galvanometer. 

205  148  Thomson  galvanometers. 

207  149  Shunt. 

208  150  Measurement  of  an  instantaneous  discharge. 
210  151  Solenoid.     Cylindrical  bobbin. 

214  152     Electrodynamometer. 

215  153     Case  of  a  ring-shaped  bobbin  or  solenoid. 

ELECTROMAGNETIC    ROTATIONS   AND    DISPLACEMENTS. 

218       154     General  statements. 

218       155     Rotation  of  a  current  by  a  magnet. 


X  CONTENTS, 

PAGE   SECTION 

219  156  Rotation  of  a  brush-discharge. 

220  157  Barlow's  wheel. 

221  158  Rotation  produced  by  reversing  a  current. 

222  159  Mutual  action  of  currents. 

222  160     Reaction  produced  in  a  circuit  traversed  by  a  current. 

223  161     Explanation  of  electromagnetic  displacements  based  on  the 

properties  of  lines  of  force. 
• 

ELECTROMAGNET.;. 

226  162     Description  and  definitions. 

227  163     Energy  expended  in  electromagnets.     General  definition  of 

the  coefficient  of  self-induction  of  a  circuit. 

230       164     Magnetic  circuit.     Magnetomotive  force.     Magnetic  resist- 
ance or  reluctance. 

234       165     Forms  and  construction  of  electromagnets. 

240       166     Magnetization  of  a  conductor. 

240  167     Modifications  in  the  properties  of  bodies  in  a  magnetic  field. 

241  168     Hall  effect. 

CHAPTER  VI.     Units  and  Dimensions. 

244  169     Units. 

245  170     Dimensions. 

247  171     C.  G.  S.  system  of  units. 

ELECTRICAL   AND   MAGNETIC    UNITS. 

248  172     General  considerations. 

249  173     Systems  in  terms  of  K and  jit. 
252       174     Proposed  nomenclature. 

254       175     Dimensions  of  K  and  ju. 

256  176     Value  of  ratio  "  z>." 

257  177     Practical  system. 

259  178     Nomenclature  of  practical  units. 

260  179     "Rational"  system. 

263       180     Electrical  standards  of  measure. 

265  .    181     Recommendations  of  Congress  in  1893. 

CHAPTER  VII.     Electromagnetic  Induction. 

266  182     Induced  currents. 

268  183  Lenz's  law. 

269  184  General  law  of  induction. 

270  185  Maxwell's  rule. 

271  186  Faraday's  rule. 

272  187  Seat  of  the  electromotive  force  of  induction. 


CONTENTS.  XI 

PAGE    SECTION 

273       188     Flux  of  force- producing  induction. 

277  189     Quantity  of  induced  electricity. 

APPLICATIONS    OF    THE    LAWS    OF    INDUCTION. 
» 

278  190     Movable  conductor  iniva  uniform  field. 

278  191     Faraday's  disc. 

279  192     Measurement  of   the  intensity  of  the  magnetic  field  by  the 

quantity  of  electricity  induced. 

280  193     Expression  for  the  work  absorbed  in  magnetization.     Loss 

due  to  hysteresis. 
283       194     Self-induction   in  a   circuit   composed  of  linear  conductors. 

Case  of  a  constant  electromotive  force.      Time  constant. 
287       195     Work  accomplished  during  the  variable  period. 

289  196     Application  to  the  case  of  derived  currents. 

290  197     Discharge  of  a  condenser  into  a  galvanometer  with  shunt. 
292       198     Self-induction  in  a  circuit  of   linear  conductors  where  there 

is  a  periodic  or  undulatory  electromotive  force. 
298       199     Graphic  representations. 
301       200     Mean    current   and  effective    current.     Measurement   by   a 

dynamometer. 
305       201     Mutual  induction  of  two  circuits. 

305  202     Mutual  induction  of  two  fixed  circuits. 

306  203     Quantity  of  induced  electricity. 

307  204     Expression  for  mutual  inductance. 

308  205     Induction  in  metallic  masses. 

309  206     Foucault  currents. 

310  207     Cores    of   electromagnets    traversed    by   variable   currents. 

Calculation  of  the  power  lost  in  Foucault  currents. 

312  208  Self-induction  in  the  mass  of  a  cylindrical  conductor.  Ex- 
pression for  the  coefficient  of  self-induction  in  such  a 
conductor. 

ROTATIONS    UNDER   THE  ACTION  OF   INDUCED    CURRENTS. 

317  209     General  rule. 

318  210     Ferraris'  arrangement. 

320       211     Shallenberger's  arrangement. 

320  212  Repulsion  exercised  by  an  inducing  current  upon  an  induced 
current. 

CHAPTER  VIII.     Impedance. 

324  213  Inductance. 

327  214  Inductance  and  capacitance. 

328  215  Inductance  and  reactance. 
331  216  Joint  impedance. 


Xll  CONTENTS. 

CHAPTER  IX.     The  Propagation  of  Currents. 

PAGE    SECTION 

333       217     Phenomena  which  accompany  the  propagation  of  the  current 

in  a  conductor. 
337       218     Special  characteristics  shown  by  alternating  currents. 

339  219     Comparative  effects  of  the  self-induction  and  capacity  of  a 

circuit. 

340  220     Effect   of   a   capacity  in  a  circuit  traversed  by  alternating 

currents. 

342       221     Combined  effects  of  a  capacity  and  a  self-induction  in  a  cir- 
cuit traversed  by  alternating  currents.      Ferranti  effect. 

344       222     Oscillating  discharge. 

350       223     Transmission  of  electric  waves  in  the  surrounding  medium. 

357       224     Present  views  on  the  propagation  of  electric  energy. 

CHAPTER  X.     Electrical  Measurements. 

361       225     Case  of  a  continuous  current. 

361  226     Siemens  Wattmeter. 

362  227     Case  of  a  periodic  current. 

362       228     Non-inductive  conductors.  .     . 

364       229     Conductors  having  self-induction. 

MEASUREMENT    OF   THE   INTENSITY    OF  A   MAGNETIC  FIELD. 

367       230     Method  by  oscillation. 

367  231     Electromagnetic  method. 

368  232     Method  based  on  induction. 

MEASUREMENT    OF    MAGNETIC    PERMEABILITY. 

369  233     Method  based  on  induction. 

371  234  Another  similar  method, 

372  235  Remarks  on  these  methods. 

373  236  Magnetometric  method. 

374  237  Method  by  portative  power. 

MEASUREMENT    OF    COEFFICIENTS    OF  INDUCTION. 

375  238     Maxwell  and  Rayleigh's  method  for  measuring  a  coefficient 

of  self-induction  in  terms  of  a  resistance. 

377  239     Maxwell's  method,  modified  by  Pirani,  for  measuring  a  self- 

induction  in  terms  of  a  capacity. 

378  240     Ayrton    and    Perry's    method    for  comparing    a    coil's  self- 

induction  with  that  of  a  standard  coil. 
380       241     Mutual  induction — Carey-Foster  method. 


THERMOELECTRIC  COUPLES. 


INTRODUCTION. 

UNITS   OF   MEASUREMENT. 

A  phenomenon  is  well  known  only  when  it  is  possi- 
ble to  express  it  in  numbers.      (KELVIN.) 

I.  Fundamental  Units. — All  electrical  actions  are  re- 
ferred to  forces,  and  are  consequently  expressed  by  the  aid 
of  the  three  fundamental  quantities,  length,  mass,  and  time. 

To  measure  these  quantities  electricians  have  chosen  the 
centimetre,  the  gram,  and  the  second  as  units. 

The  centimetre  is  approximately  the  billionth  part  of  the 
terrestrial  quadrant ;  rigorously  speaking,  it  is  the  hundredth 
part  of  the  standard  metre  measured  by  Delambre  and 
Borda,  and  kept  at  the  international  conservatory  at  Sevres. 

The  gram  represents  about  the  mass  of  a  cubic  centi- 
metre of  distilled  water  at  its  maximum  density.  There  is 
also  a  standard  kilogram  at  Sevres. 

The  second  is  the  86,4OOth  part  of  the  mean  solar  day. 

These  units,  called  fundamental,  are  represented  by  the 
symbols  [£],  [J/],  [T]. 

The  numerical  value  of  a  quantity  is  expressed  by  its  ratio 
to  the  unit  chosen.  A  length  measured  by  a  number  /  will 
have  a  concrete  value  equal  to  /[£].  If  we  adopt  another 
unit  [Z/],  there  will  be  a  numerical  value  /',  such  that 

/'[//]  =  /[£],    whence     ljt  =         . 


2  IN  TROD  UCTION. 

We  see,  therefore,  that  the  numerical  value  of  a  quantity 
is  in  inverse  ratio  to  the  magnitude  of  the  unit  chosen. 

2.  Derived    Units. — In    order   to    express   the    various 
physical  quantities,  arbitrary  units  might   be  chosen  quite 
independent  of  each  other.     This  method,  which  was  long 
followed,  presents  no  inconvenience  when  the  measures  are 
relative,  that  is,  when  they  are  directly  compared  with  their 
units.     But  more  often   quantities  are  measured  by  units  of 
other  kinds,  making  use  of  the  inter-relations  between  the 
different  quantities.     Such  a  system  of  measurement  is  called 
absolute.     For  example,  to  measure  a  surface  we  do  not  com- 
pare it  directly  with  a  standard  area,  but  determine  its  linear 
elements,  by  aid  of  the  unit   of  length,  and  then  apply  the 
relation  existing  between  an  area  and  its  linear  dimensions. 

For  a  square  s,  with  a  side  /,  the  relation  is  s  —  kP.  If 
/=  i,  j  =  k. 

The  arbitrary  factor  k,  which  represents  the  area  of  a 
square  having  unit  side,  may  be  put  equal  to  i.  The  unit 
of  area  thus  determined  is  the  square  whose  sides  are  equal 
to  one  centimetre  ;  it  is  connected  with  one  of  the  funda- 
mental units,  and  for  this  reason  is  called  the  derived  unit  of 
area. 

In  the  same  way  the  derived  unit  of  volume  is  the  cube 
having  sides  of  one  centimetre. 

We  can  thus  define  derived  units  for  all  the  physical  mag- 
nitudes, getting  rid  of  the  arbitrary  coefficients  in  the  rela- 
tions which  unite  these  magnitudes  together. 

The  system  of  units  determined  in  this  way  is  called  by 
the  initials  of  the  fundamental  units  chosen. 

The  C.  G.  S.  system  of  units,  adopted  by  electricians,  has 
as  its  basis  the  centimetre,  the  gram,  and  the  second. 

3.  Example  of  a  Derived  Unit. — The  velocity  of  a  mov- 


UNI  TS  OF  ME  A  S  U  RE  MEN  T.  3 

ing  body,  traversing  a  path  /  in  a  time  ^,  is  given  by  the 
equation 

' 


k  expresses  the  velocity  of  %  moving  body  traversing  unit 
length  in  unit  time.  This  velocity  is  chosen  as  unity,  thus 
eliminating  a  factor  inconvenient  in  calculation.  This  unit 
is  the  velocity  of  one  centimetre  per  second',  it  may  be  ex- 
pressed in  symbolic  form 


4.  Dimensions  of  a  Derived  Unit.  —  Such  an  expression, 
which  shows  the   dependence  of  the  derived   unit  on   the 
fundamental   units,   exhibits  the   dimensions  of  the  derived 
unit.     It  enables  us  to  follow  the  variation   of  the  derived 
unit  when  the  fundamental  units  are  changed.     If,  for  ex- 
ample, we   measure   the  time  in   hours   and  the   length  in 
metres,  the  derived  unit  of  velocity  will  be  \L'T'  ~  ']  =  100 
X  3600  ^[ZjT-1],  or  the  thirty-sixth  part  of  the  unit  defined 
above. 

Every  relation  between  physical  quantities  is  independent 
of  the  units  chosen  to  measure  it,  therefore  this  relation 
must  be  homogeneous  with  regard  to  the  fundamental 
units. 

Thus  the  equation  v  —  al,  in  which  v  represents  a  veloc- 
ity, /  a  length,  and  a  an  abstract  number,  is  inconsistent, 
for  only  the  first  member  would  vary  with  the  unit  of  time. 

5.  Mechanical   Derived   Units.  —  Following  the  line  of 
reasoning  used  above,  it  is  readily  seen  that  the  unit  of  an- 
gular velocity  is  the  velocity  of  a  moving  body  which  passes 
over  unit  angle  in    unit    time.     As  the  unit  angle  or  radian 
(arc  equal    to  the  radius)  is  defined  by  a  simple   numerical 
ratio,  the  dimensions  of  angular  velocity  reduce  to  \T~  ']. 


4  IN  TROD  UCTION. 

The  unit  of  acceleration  is  the  acceleration  by  which  the 
velocity  is  increased  by  one  unit  per  second.  Dimensions 
[LT-1. 

The  unit  of  quantity  of  movement  (momentum)  is  the  mo- 
mentum of  unit  mass  moving  with  unit  velocity.  Dimen- 
sions \LMT  " ']. 

The  unit  of  force,  which  has  received  the  name  dyne,  is 
the  force  which,  applied  to  unit  mass,  impresses  upon  it  unit 
acceleration.  Dimensions  \LMT~*\ 

The  ordinary  unit  of  force  is  the  weight  of  the  gram, 
that  is,  the  force  capable,  in  our  latitude,*  of  impressing  on 
unit  mass  an  acceleration  approximately  equal  to  981  cm. 
per  second.  The  gram  is  consequently  equal  to  981 
dynes. 

The  unit  of  work,  called  erg,  is  the  work  done  by  unit 
force  on  a  body  moving  in  the  direction  of  its  action  over 
unit  length.  Dimensions  \_UMT-*]. 

The  ordinary  unit  of  work  is  the  kilogrammetre,  which  is 
equal  to  981  X  IO5  ergs,  in  our  latitude.* 

The  unit  of  po^ver,  or  erg  per  second,  is  the  power  de- 
veloped when  unit  work  is  done  in  unit  time.  Dimensions 
[/,2 MT  ~ ']. 

The  ordinary  units  of  power  are  the  cheval-vapeur  (French 
horse-power),  which  is  equal  to  75  X  981  X  IO5  =  736  X  io7 
ergs  per  second,  and  the  poncelet,  defined  by  the  Congress  in 
Mechanics,  1889,  as  equivalent  to  100  kilogrammetres  per 
second,  or  981  X  io7  ergs  per  second. 

The  unit  of  density  is  the  density  of  a  body  which  con- 
tains unit  mass  in  unit  volume.  Dimensions  [L~3M], 

The  unit  of  modulus  of  elasticity  is  the  modulus  of  a  body 
which,  supporting  unit  force  per  unit  of  section,  receives 


*  Li&ge,  lat.  50°  45'  approx. 


UNITS  OF  MEASUREMENT.  $ 

an    elongation    equal   to    its   original  length.      Dimensions 


6.  Principle  of  the  Conservation  of  Energy. — Work  ap- 
plied to  a  system  is  capable^of  various  effects.  It  may  be 
used  :  (i)  To  increase  the  active  energy  of  the  masses,  or,  to 
use  Rankine's  expression,  to  develop  kinetic  energy,  repre- 
sented by  the  product  of  half  the  sum  of  the  masses  into  the 
square  of  their  velocity.  (2)  To  overcome  the  friction  of 
the  system  ;  it  was  long  believed  that  this  effect  represented 
a  loss  of  energy,  but  thermodynamics  has  shown  that  in 
such  a  case  there  is  generated  an  amount  of  heat  equivalent 
to  the  work  expended.  (3)  To  overcome  molecular  forces, 
such  as  elasticity,  chemical  affinity  ;  or  to  overcome  natural 
forces,  such  as  gravitation,  magnetic  attraction,  etc.  In  this 
case  the  work  is  stored  up  in  the  system  in  the  form  of  po- 
tential energy,  which  is  again  transformed  into  kinetic  energy 
or  heat,  when  the  system  is  abandoned  to  the  reaction  of 
the  forces  concerned. 

Let  us  suppose,  for  example,  that  we  raise  a  weight  or 
stretch  a  spring  which  sets  a  clockwork  in  motion.  The 
potential  energy  given  to  the  weight  or  spring  is  trans- 
formed into  kinetic  energy  when  the  mechanism  is  allowed 
to  operate,  and  this  kinetic  energy  is  itself  reduced  to  heat 
by  the  friction  of  the  wheelwork. 

The  tendency  of  modern  science  is  to  refer  these  diverse 
varieties  of  energy  to  a  single  one,  kinetic  energy  ;  calorific, 
luminous,  or  electrical  radiations,  for  example,  which  to  us 
seem  potential  forms  of  energy,  might  be  reduced  to  special 
modes  of  motion  of  the  ether. 

The  study  of  physical  phenomena  has  given  us  a  natural 
law  of  the  highest  importance.  The  energy  of  a  system  is  a 
quantity  which  cannot  be  either  increased  or  diminished  by  any 
mutual  action  between  the  bodies  which  compose  the  system* 


IN  TROD  UCTION. 

This  law  of  the  conservation  of  energy,  together  with  that  of 
the  conservation  of  matter,  rules  supreme  in  physical  science. 

From  this  principle  it  results  that  a  system  cannot  of  it- 
self produce  more  than  a  limited  quantity  of  external  work, 
whence  the  impossibility  of  perpetual  motion. 

The  persistence  and  the  indestructibility  of  energy  make 
it  as  much  a  physical  entity  as  matter  is,  and  give  it  a 
leading  place  among  the  magnitudes  considered  in  me- 
chanics. Energy  assumes  indifferently  a  mechanical,  elec- 
trical, thermic,  or  chemical  form.  Experiment  shows  that 
the  two  former  are  capable  of  being  entirely  transformed 
into  one  of  the  two  latter,  but  that  only  a  portion  of  ther- 
mic or  chemical  energy  can  be  made  to  assume  a  mechan- 
ical or  electrical  form. 

In  whatever  form  energy  may  be,  it  possesses  a  mechani- 
cal equivalent ;  it  is  therefore  homogeneous  with  work 
\UMT~*~\)  and  maybe  measured  in  mechanical  units.  It 
follows  that  the  C.  G.  S.  unit  of  heat  equals  the  erg. 

One  gram-degree,  or  lesser  calorie  (caloriegram),  rep- 
resents 4.2  X  io7  C.  G.  S.  units  of  heat. 

The  electrician  has  constantly  occasion  to  apply  the 
principle  of  the  conservation  of  energy,  which  we  have  just 
defined,  for  the  essential  role  of  electricity  is  to  serve  as 
agent  for  the  transformation  of  energy.  The  energy  of  the 
electric  current  is  produced  from  the  work  done  by  chemi- 
cal affinity  in  batteries,  by  using  up  heat  in  thermo-electric 
couples,  or  by  an  absorption  of  mechanical  power  in 
dynamos. 

The  energy  of  the  current  is,  in  its  turn,  transformed  into 
heat  and  light  in  the  conductors  and  electric  lamps  ;  it  is 
capable  of  decomposing  an  electrolyte  or  of  overcoming  the 
resistance  offered  to  the  motion  of  an  electromotor. 

The  marvellous  facility  with  which  electricity  lends  itself 
to  the  transmission  and  transformation  of  energy,  and  which 


UNITS   OF  MEASUREMENT.  7 

justifies  the  increasing  number  of  applications  of  this  agent, 
leads  the  electrician  to  compare  phenomena  of  very  diverse 
kinds,  the  measurement  of  which  demands  such  a  system  as 
the  C.  G.  S.,  embracing  all. physical  magnitudes. 

7.  Multiples  and  Submultiples  of  the  Units. — The  use 

of  the  units  just  described  leads  sometimes  to  very  large  or 
very  small  numerical  values.  By  way  of  abbreviation  we 
make  use  of  multiples  or  submultiples  designated  by  such 
prefixes  as  kilo-,  mega-  (one  million),  milli-,  micro-  (one 
millionth). 

Thus,  one  megadyne  =  10"  dynes ; 
one  microdyne  =  io~6  dynes. 

8.  Application  of  the  Dimensions  of  Units. — The    di- 
mensions of  the  units  are  of  use  not  only  in  verifying  the 
homogeneity  of   formulas,  but   they  allow  us,  as    Bertrand 
has  shown,  to  predict  the  form  of  a  function  when  the  physical 
quantities  which  enter  into  it  are  known.     Suppose,  for  ex- 
ample, that  experiment  has  shown  that  the  velocity  of  prop- 
agation of  an  undulatory  movement  in  a  medium  depends  on 
the  modulus  of  elasticity  and  the  density  of  the  medium. 

Then  the  velocity  v  is  a  function  of  the  elasticity  e,  and 
the  density  d\ 

v  =  <f>(e,  d). 

If  we  consider  the  dimensions  of  the  quantities  which 
enter  into  this  equation,  we  have 

vLT~l  =  <t>(eL-lMT-\  dL~*M). 

As  Mis  wanting  in  the  first  member,  the  homogeneity  of 
the  function  requires  it  to  be  eliminated  from  the  second, 
which  is  obtained  by  adopting  the  form 


8  IN  TROD  UCTION. 

To  bring  L  and  T  to  the  same  degree  in  both  members, 
it  is  clear  that  the  function  must  be  a  radical  of  the  second 
degree.  From  what  precedes  we  conclude  that  v  is  a  linear 
function  of 


And  in  fact  experiment  shows  that  the  relation  sought  is 


v  — 


GENERAL  THEOREMS   RELATIVE   TO   CENTRAL  FORCES. 

9.  Definitions. — Forces  are  called  central  whose  direction 
passes  through  definite  points,  called  centres  of  force,  and 
whose  intensity  is  a  function  of  the  distance  between  those 
points. 

The  Newtonian  central  forces,  such  as  gravitation,  electric 
and  magnetic  attraction,  are  inversely  proportional  to  the 
square  of  the  distance  between  the  acting  centres. 

In  studying  the  effects  of  these  forces,  it  is  a  matter  of 
indifference  whether  they  emanate  from  the  centres  them- 
selves or  have  their  seat  in  the  medium  which  separates 
these  centres.  Thus,  to  account  for  the  universal  attrac- 
tion of  matter,  the  simplest  way  is  to  assume  that  the 
attractive  force  is  a  property  of  all  ponderable  bodies,  which 
act  upon  each  other  at  a  distance.  This  hypothesis  has 
the  advantage  of  lending  itself  readily  to  calculation.  It 
has  sufficed  as  a  basis  for  celestial  mechanics. 

Nevertheless  it  does  not  satisfy  the  intellect.  The  ordi- 
nary methods  used  in  the  transmission  of  forces  show  us  the 
necessity  of  an  intermediary,  such  as  a  tense  cord,  air  or 
water  under  pressure,  and  this  permits  us  at  least  to  limit  to 


GENERAL  THEOREMS  RELATIVE   TO  CENTRAL  FORCES.   9 

intermolecular  space  the  idea  of  action  at  a  distance. 
Again,  the  direct  action  of  one  body  on  another  takes  for 
granted  an  instantaneous  effect.  Now  physical  phenomena, 
even  the  most  rapid,  have  p.  finite  time  of  propagation. 

To  account  for  observed  phenomena,  physicists  have  been 
led  to  suppose  the  'universe  filled  with  an  ocean  of  ether, 
whose  waves,  representing  heat,  light,  and  electrical  energy, 
are  propagated  with  a  velocity  of  3  X  IO10  centimetres  per 
second,  so  that  they  take  about  eight  minutes  to  reach  us 
from  the  sun. 

However,  for  simplicity  of  treatment,  we  shall  admit  pro- 
visionally that  central  forces  are  due  to  the  bodies  from 
which  they  seem  to  emanate,  or  to  an  agent  diffused  through 
these  bodies. 

In  the  case  of  gravity  the  observed  actions  are  attributed 
to  the  mass  of  the  body. 

In  the  case  of  the  electrical  phenomena  that  are  mani- 
fested between  bodies  that  have  been  rubbed,  we  shall 
say  that  an  agent,  called  electricity,  has  been  developed  on 
these  bodies,  and,  without  making  any  supposition  as  to  its 
nature,  we  shall  speak  of  quantity,  mass,  or  charge  of  the 
agent,  these  terms  expressing  merely  a  factor  proportional 
to  the  effects  produced. 

Thus  we  shall  say  that  two  bodies  possess  equal  quantities 
of  the  agent  when  they  produce  equal  effects  on  a  third 
body.  The  quantities  of  the  agent  will  be  doubled,  or 
tripled,  when  the  forces  developed  are  double  or  triple. 

The  quantity  of  agent  per  unit  of  area  or  per  unit  of 
volume  is  called,  the  surface  density  or  volume  density. 

10.  Elementary  Law  Governing  the  Newtonian  Forces. 

• — The  preceding  definitions  amount  to  saying  that  the  force 
exerted  between  two  quantities  of  the  agent  is  proportional 
to  the  product  of  these  quantities,  since  it  is  proportional 


I  O  IN  TROD  UCTION. 

to  each  one  of  them.  It  is  also  a  function  of  the  distance 
between  the  masses  concerned.  In  the  case  of  Newtonian 
forces  it  is  inversely  proportional  to  the  square  of  the  dis- 
tance. 

If,  then,  we  express  by  m,  m'  two  quantities  of  the  agent, 
and  by  /  their  distance,  the  force 


. 

The   action    exerted  on  one  of  the  masses   considered    as 
unity  would  be  expressed  by 

H=kj- 

In  the  case  of  electric  and  magnetic  actions,  masses  of  the 
same  nature  repel  each  other,  contrary  to  what  holds  in  case 
of  gravitation. 

In  a  logical  system  of  units,  the  constant  k  is  not  a  simple 
numerical  factor.  Consider  the  attraction  of  heavy  bodies 
and  replace  force,  mass,  and  distance  by  their  dimensions  : 
then  the  condition  of  homogeneity  demands  that  k  have 
dimensions  \DM-*T-*]. 

II.  Field  of  Force.  —  Let  us  suppose  that  the  quantities 
m,  m',  m"  of  the  agent  are  concentrated  in  physical  points, 
occupying  given  positions  in  space.  If  we  bring  into  their 
vicinity  a  mass  of  the  agent  equal  to  unity,  it  is  acted  upon 
by  the  forces  emanating  from  m,  m',  m",  which  form  a  result- 
ant having  a  definite  direction  and  intensity.  By  changing 
the  position  of  the  point  charged  with  unit  mass  of  the 
agent  we  can  obtain  the  intensity  and  sign  of  the  resultant 
force  for  every  point  in  space. 

The  space  in  which  such  forces  are  manifested  is  called  a 
field  of  force,  and  the  resultant  force,  just  defined,  is  the 
intensity  of  the  field  at  the  point  where  the  unit  mass  is 


GENERAL  THEOREMS  RELA  TIVE  TO  CENTRAL  FORCES.    1  1 

placed.  The  direction  of  the  resultant  is  called  indirection 
of  the  field. 

The  value  of  the  intensity  of  a  field  is  directly  deduced  by 
the  application  of  the  elem-entary  law  ;  but  the  necessary 
calculations  by  this  method  ,  become  extremely  complicated 
in  the  case  of  a  number  of  acting  masses,  since  the  elements 
to  be  combined  are  vectors,  that  is,  quantities  having  given 
magnitudes  and  directions,  and  combining  according  to  the 
parallelogram  of  forces.  The  procedure  is  especially  in- 
volved when  the  analytical  method  is  employed. 

The  solution  of  the  problem  is  reduced  to  a  simple  alge- 
braic addition  followed  by  a  differentiation,  by  taking  into 
consideration  a  new  function  defined  by  Laplace  and  inves- 
tigated by  Gauss  and  Green  under  the  nz.mz  potential. 


12.  Potential.  —  Let  us  suppose  that  unit  mass   is   dis- 


FlG.    I. 

placed  in  the  field  by  an  infinitesimal  distance,  under  the 
action  of  the  forces  acting  upon  it. 

Let  oo'  =  dr  be  the  displacement.      The  force  due  to  the 

mass  m  is  -rr,  and  the  work  done  under  the  action  of  this 
force  is 

km  ,  km 

—  dr  cos  a  = 


1 2  IN  TROD  UCTION. 

Likewise  the  work  done  by  m'  is 

km'  jf 

by  «", 

~^r. 

These  single  expressions  for  the  work  done  are  to  be 
added,  since  they  are  taken  in  the  same  direction ;  the  total 
work  is  therefore  expressed  by 

,mdt 


This  sum  is  the  differential  of  the  function 

m 
—  k*2~r  -\-  const. 

The  expression -^-k'S—,  whose  differential,  taken  with  the 

contrary  sign,  represents  the  elementary  work  of  the  forces 
of  the  field,  has  been  given  the  name  oi  potential  by  Gauss. 
We  shall  designate  it  by  the  letter  U: 


=  . 

For  a  point  in  space,  therefore,  the  potential  is  proportional 
to  the  sum  of  the  ratios  of  the  acting  masses  to  their  distances 
from  the  point. 

The  potential  permits  us  readily  to  define  the  work  ac- 
complished by  the  forces  of  the  field. 

Thus,  if  we  integrate  the  expression 


between  two  positions  (9,,  <92,  occupied  by  the  unit  mass, 
we  get 


- 

o,    l 


GENERA  L  THEOREMS  RELA  TIVE  TO  CENTRAL  FORCES.    1  3 

The  work  done  by  the  field  on  unit  mass  displaced  from  the 
point  Ol  to  <92  is  equal  to  the  difference  of  the  values  of  the 
function  U  at  the  two  points. 

The  work  depends  solely,  upon  the  position  of  the  initial 
and  final  points,  and  not  on  the  path  followed  by  the  unit 
mass  between  these  points. 

If  the  unit  mass  should  pass  from  the  point  O1  to  an  in- 
finite distance  from  the  acting  masses,  we  would  have 


r°  A/      r* 

/     -7T  =   /      -dU  =  U,. 

J  ol  J  ol 


Hence  we  see  that  the  potential  at  any  point  is  measured 
by  the  work  done  by  the  field  in  displacing  unit  mass  from 
the  given  point  to  an  infinite  distance  from  the  acting  masses, 
that  is,  to  the  limit  of  the  field. 

The  potential  function  furnishes  a  simple  expression  for 
the  intensity  of  the  field. 

Let  H  be  the  component  of  intensity  in  a  direction  /. 
The  elementary  work  //d/  is  likewise  expressed  by  the  dif- 
ferential, taken  with  contrary  sign,  of  the  potential  in  this 
direction: 

ffdl=  - 
whence 


The  component  of  field  intensity  in  a  given  direction  is  ex- 
pressed  by  the  derivative,  taken  with  contrary  sign,  of  the  po- 
tential in  that  direction. 

The  force  is  directed  towards  the  points  where  the  poten- 
tial diminishes. 


14  IN  TROD  UCTION. 

13.  Equipotential  Surfaces.  —  Put 

U  =  </>(#,  y,  z)  =  constant, 

x,  y,  and  z  representing  the  points  of  the  field  by  rectangular 
co-ordinates. 

This  equation  represents  a  surface  at  every  point  of  which 
the  potential  has  the  same  value.  Consequently  the  forces 
of  the  field  have  a  zero  resultant  along  this  surface,  the 
normal  to  which  represents  the  direction  of  the  field  at  each 
point. 

The  surfaces  thus  defined  are  called  equipotential  surfaces 
or  level  surfaces,  by  analogy  with  the  free  surface  of  a 
liquid,  everywhere  normal  to  the  force  of  gravity. 

Designating  by  n  a  direction  normal  to  the  equipotential 
surface,  the  field  intensity  in  a  point  of  the  surface  is  ex- 
pressed by 


We  can  get  a  representation  of  the  distribution  of  the 
forces  of  the  field  by  imagining  in  the  field  a  series  of 
similar  surfaces  sufficiently  near  to  each  other  and  corre- 
sponding to  potentials  which  increase  in  arithmetical  pro- 
gression. 

A  mass  free  to  move  in  the  field  will  follow  a  path  cutting 
the  equipotential  surfaces  perpendicularly.  This  curve, 
whose  tangent  represents  in  each  point  the  direction  of  the 
field,  has  been  named  by  Faraday  a  line  of  force. 

The  field  intensity  is  obviously  in  inverse  ratio  to  the  seg- 
ment of  line  of  force  comprised  between  two  consecutive 
equipotential  surfaces. 

14.  Case  of  a  Single  Mass.  —  The  case  of  a  single  acting 


GENERAL  THEOREMS  RELA  TIVE  TO  CENTRAL  FORCES.    I  5 

mass  gives  an  example  of  a  field  easily  defined.  The  equi- 
potential  surfaces  are  concentric  spheres,  whose  radii  repre- 
sent the  lines  of  force. 

Let  us  suppose  a  mass  m,  such  that  km  =  6,  concentrated 
in  a  point  A. 

The  concentric  circles  represent  the  intersection  by  a  plane 
passing  through  the  mass  m  of  the  equipotential  surfaces  I, 
2,  3,  4,  5,  6.  The  radii  of  these  circumferences  are  respec- 
tively 6/1,  6/2,  6/3,  6/4,  6/5,  6/6. 

15.  Uniform  Field. — We  see  that  as  the  potential  de- 
creases the  equipotential  surfaces  are  successively  further 
and  further  apart.  At  a  sufficiently  great  distance  from  the 


FIG.  2. 

centre  the  lines  of  force  drawn  through  a  region  of  small  ex- 
tent are  practically  parallel,  and  the  equipotential  surfaces 
are  comparable  to  planes  in  this  region.  In  the  case  of 
gravitation,  for  example,  no  appreciable  error  is  caused  by 
taking,  in  the  space  occupied  by  a  laboratory,  the  verticals 
as  parallel. 

A  field  represented  in  this  manner  by  equipotential  planes 
and  lines  of  force  perpendicular  to  them,  whose  intensity  is 
constant  in  magnitude  and  direction,  is  called  a  uniform  field. 

16.  Case  of  Two  Acting  Masses. — Let  us  consider  the 


i6 


INTRODUCTION. 


case  of  two  acting  masses,  such  that  for  one  of  them  km  = 
20,  and  for  the  other  km'  =  5. 

To  determine  the  intersection  of  the  equipotential  sur- 
faces due  to  the  two  centres  by  a  plane  passing  through 
these  centres,  commence  by  tracing  the  circular  equipoten- 
tial lines  due  to  each  centre  considered  separately. 

Let  nt,  n^,  nz,  nt  ...  be  the  circles  drawn  around  the  first, 
and  «/,  #„',  #,'  .  .  .  those  enveloping  the  second. 


FIG.  3. 

The  equipotential  line  of  the  order  5  will  evidently  pass 
through  the  intersections  of  the  circumferences  nt,  «/ ;  «,, 

*,';  «,,  «,';  »i,  »/• 

The  equipotential  line  of  the  order  4  will   pass  through 


GENERAL  THEOREMS  RELA  TIVE  TO  CENTRAL  FORCES.    I/ 

the  intersections  of  the  circumferences  na,  n/ ;  n.lt  »/ ;  «,, 
«,',  and  so  on  for  the  other  orders. 

The  lines  of  force  will  be  curves  normal  to  the  equipoten- 
tial  lines  obtained.  > 

In  Fig.  3,  taken  from  Maxwell's  Electricity  and  Magnetism, 
the  two  masses  above  mentioned  are  concentrated  in  the 
points  A  and  B.  The  full  lines  are  equipotential ;  the  lines 
of  force  are  shown  by  dotted  lines. 

17.  Tubes  of  Force. — Trace  any  closed  curve  in  a  field, 
and  imagine  that  a  line  of  force  passes  through  each  point 
of  this  curve.     All  these  lines  taken  together  form  a  tubular 
surface,    called    a  tube  of  force.     In  the  case  of   a   single 
centre  of  force  the  tubes  of  force  are  conical.    In  a  uniform 
field  they  are  cylindrical. 

18.  Flux  of  Force. — The  intensity  of  any  field   is  con- 
stant over  an  infinitely  small  surface  dj.     The  product  of 
this  surface  into  the  component  of  the  intensity  normal  to 
the  surface  is  called  \\\z  flux  of  force  across  the  surface. 

Let  a  be  the  angle  of  the  direction  of  the  field  with  the 
normal,  the  flux  of  force  will  be  represented  by 

dN  =  H  cos  a  ds. 
The  flux  of  force  across  a  finite  surface  is  given  by 

N  —j  H  cos  ads, 

the  integration  being  extended  to  every  element  of  the  sur- 
face under  consideration. 

In  the  case  of  a  closed  surface  the  flux  is  said  to  be  issuing 
when  the  lines  of  force  are  directed  towards  the  exterior  of 
the  surface,  and  entering  in  the  opposite  case. 

By  considering  the  angle  a  to  be  made  by  the  direction  of 
the  field  with  the  normal  exterior  to  the  surface,  the  change 
of  sign  of  cos  a  allows  us  to  distinguish  the  issuing  from  the 
entering  flux. 


1 8  IN  TROD  UCTION. 

19.  Theorem. —  The  flux  of  force  which  traverses  a  tube 
of  infinitely  small  section  is  independent  of  the  inclination  of 
the  section  to  the  axis  of  the  tube. 

Thus,  &H  =  H  cos  ads  =  #dcr, 

dcr  representing  the  section  of  the  tube  normal  to  the  axis, 
and  H  the  intensity  of  the  field  at  this  point. 

20.  Gauss'  Theorem. — Between  the  masses  of  a  field  and 
the  flux  traversing  a  surface  which   envelops  these  masses 
there  exists  a  simple  relation,  very  frequently  used,  as  fol- 
lows: 

The  flux  of  force  traversing  a  closed  surface  in  a  field  is 
equal  to  ^.nk  times  the  sum  of  the  masses  enveloped  by  this  sur- 
face. 

I.  Let  us  first  consider  a  single  mass  m,  concentrated  in 


FIG.  4. 

a  point  P,  within  a  surface  which,  to  make  our  treatment 
more  general,  has  a  re-entrant  portion  (Fig.  4). 

From  the  point  P  as  apex  draw  the  elements  of  a  cone 
corresponding  to  a  solid  angle  do?,  which  is  measured  by 
the  surface  intercepted  by  the  cone  on  a  sphere  of  centre  P 
and  radius  equal  to  unity. 

Call  ds,  d.y',  d/'  the  areas  bounded  by  the  intersections  of 
the  cone  with  the  surface  described;  do?  represents  the 
apparent  surface  of  these  intersections  as  seen  from  the 
point  P.  Call  /,  /',  I"  the  distances  of  the  intersected  ele- 


GENERAL  THEOREMS  RELA  TIVE  TO  CENTRAL  FORCES,    ig 

ments  from  the  point  P\  and   a,  a',  a"  the  angles  of  the 
axis  of  the  cone  with  the  normals  to  the  elements. 

The  flux  of  force  traversing  these  elements  are  respectively 

i    km  ,    km  ,,  ,         .    km  ,  ,, 

+  —  cos  ads,      +  -^  cosg  ds  ,      +  j^  cos  ads  • 

But 

ds  cos  a  ds'  cos  a'        ds"  cos  a" 


since  these  expressions,  by  definition,  measure  the  solid 
angle  doo.  The  flux  reduces  then  to  kmdao,  whatever  be 
the  number  of  intersections,  provided  that  the  number  is 
uneven. 

The  total  flux  across  the  surface  is  given  by  the  sum  of 
all  the  elementary  cones  that  can  be  drawn  about  P\  thus 

V 


r. 


47T  represents  the  total  surface  cut  by  the  cones  on  a 
sphere  of  unit  radius. 

II.  If  we  had  considered  a  mass  m,  outside  of  the  closed 
surface,  the  elementary  cones  can  traverse  this  surface  only 
an  even  number  of  times,  and  would  thus  give  a  zero  re- 
sultant. 

III.  Finally,  if  we  suppose  in  the  field  masses,  ml  ,  my  ,  m3  , 
some  of  which  are  in  the  interior  and  others  on  the  exterior 
of  the  surface,  the  total  flux  through  the  surface  will  be  the 
sum  of  the  flux  due  to  the  masses  within,  2m  : 


I 


Hds  cos  a  = 


21.  Corollary  I. — Suppose  that  the  closed  surface  be 
bounded  by  the  lateral  walls  of  a  tube  of  force  and  by  two 
sections  of  this  tube,  s  and  s' '. 


20  IN  TROD  UCTION. 

As  the  walls  of  the  tube  cut  no  lines  of  force,  the  flux  of 
force   traversing   the  closed  surface  is  limited   to   the    flux 

/  Hds  —  I  H'ds',  ds  and   dsf  being  normal  equipotential 
sections  of  the  tube.     Consequently, 


If  there  are  no  masses  within  this  closed  region, 


The  flux  entering  by  one  base  issues  from  the  other,  that 
is  to  say  that  the  flux  is  constant  in  a  tube  of  force,  as  long 
as  the  tube  does  not  encounter  acting  masses. 

This  property,  comparable  with  that  of  fluid  circuits  in 
which  the  flow  remains  constant  as  long  as  no  outflowing 
sources  are  met,  justifies  the  namey?&;r,  given  to  the  mathe- 
matical expression  which  we  have  been  considering.  We 
shall  see  that  this  property,  known  by  the  name  of  continuity 
of  flux,  plays  an  important  part  in  electric  and  magnetic 
phenomena. 

22.  Corollary  II.  —  If  the  tube  of  force  were  infinitely 
thin,  we  should  have 

ffds  =  Hfdsf  =  dN\ 
whence 

ILL        *L 

H     '  d/' 

In  such  a  tube  the  intensity  of  the  field  is  in  inverse  ratio 
to  the  section  normal  to  the  axis.  In  a  uniform  field  the 
tubes  of  force  are  necessarily  cylindrical. 


23.  Corollary  III.  —  The  expression  H  =  —  shows  that 

the  intensity  of  afield  is  the  flux  per  unit  equipotential  sur- 
face at  the  point  under  consideration. 


GENERAL  THEOREMS  RE  LA  T2VE  TO  CENTRAL  FORCES.    21 

24.  Unit  Tube.     Number  of  Lines  of  Force.— A  tube 
chosen  so  that  the  expression    /  Hds  =  i  is  assumed  a  unit 

tube.  ,   ' 

Following  a  convention  du^to  Faraday  and  admitted  by 
many  authors,  the  number  of  lines  of  force  of  a  field,  which 
in  reality  is  indefinite,  is  limited  to  the  number  of  unit 
tubes  of  which  they  form  the  axes. 

In  accordance  with  this  convention,  Gauss'  theorem  is 
enunciated  as  follows:  The  number  of  unit  tubes  or  lines  of 
force  traversing  a  closed  surface  in  a  field  is  equal  to  ^.nk 
times  the  sum  of  the  quantities  of  the  agent  enveloped  by 
this  surface. 

25.  Potential  Energy  of  Masses  Subjected  to  New- 
tonian Forces. — In  consequence   of  the  repulsion  exerted 
between    masses  of   the    same   kind,  a  certain    amount    of 
work   must  be  done  to  bring  the  masses  m,  m',  m"  to  the 
neighboring  points  o,  o',  o" .     This  work  is  stored  up  in  the 
system  in  the  state  of  potential  energy,  and  is  restored  when 
the  masses,  being  set  free,  separate  indefinitely  from  one 
another  under  the  effect  of  their  mutual  actions. 

To  determine  the  expression  for  the  work  done,  suppose 
that  the  masses  are  formed  by  means  of  elementary  masses 
brought  up  successively  to  the  given  points  0,  </,  oo" . 

To  bring  an  element  dm  to  a  point  o  whose  potential  is 
£7,  we  must  by  definition  do  an  amount  of  work  equal  to 
[7dm. 

For  the  other  points  we  obtain  in  the  same  way  the 
elements  of  work  U'dm',  U"dm". 

But  in  proportion  as  the  masses  increase  the  potential  of 
each  of  the  points  rises.  At  the  point  o,  for  example,  it 
passes  from  the  value  zero  to  the  value 


22  IN  TROD  UCTION. 

We  may  suppose  that  the  progressive  increase  of  the 
masses  takes  place  in  a  constant  ratio,  so  that  at  a  given 
instant  the  masses  accumulated  at  the  various  points  reach 
the  same  fractional  part  of  their  full  values.  In  these  con- 
ditions the  potentials  increase  in  the  same  ratio ;  the 

mean   value  of   the  potential   at   the    point  o  is  — ,  corre- 
sponding to  elementary  work,  —  dm. 

The  work  necessary  for  the  formation  of  the  mass  m  is 


U    f"  Urn 

=  —    I     am  =  . 

2  Jo  2 


w  =  — 


The  sum  of  the  work  expended  on  the  various  masses  of 
the  system  will  be 


W=  - 


Hence  we  see  that  the  potential  energy  of  the  system  is  the 
half  sum  of  the  products  of  the  masses  by  their  potentials. 

The  above  assumption  of  proportional  increments  of  the 
masses  does  not  in  the  least  invalidate  the  generality  of 
the  conclusion.  For  the  potential  energy  is  measured  by 
the  work  stored,  which  depends  only  on  the  final  state  of 
the  system,  and  is  in  no  way  dependent  on  the  method  by 
which  the  masses  have  come  to  this  state.  We  shall  also 
give  a  demonstration  quite  independent  of  such  hypothesis. 

When  two  masses  m,  m'  ,  at  a  distance  /,  separate  still 
further  by  a  length  d/,  the  increase  of  potential  energy  is 
equal  and  of  opposite  sign  to  the  work  accomplished.  We 
have  therefore 


j  . 

dw  =  —  k——dl. 


AP PLICA  TIONS.  23 

When  the  masses  move  to  an  infinite  distance  from  each 
other,  the  work  done  represents  their  total  initial  energy, 
that  is 


For  a  system  of  masses  we  get  an  expression  of  the 
form 

TT_       ,  _  mm' 
W  = 

Observe  that 

v  mm' 


the  factor  £  being  necessary  in  order  to  avoid  taking  each 
couple  of  quantities  twice. 

m' 
But  k*2  —  represents  the  potential  of  the  point  at  which 

the  mass  m  is  situated.     We  get,  therefore, 

J¥=~2mU. 

2 
APPLICATIONS. 

Before   going  further  let   us  apply  the  properties    just 


FIG.  5. 

demonstrated  to  some  simple  cases  which  we  shall  later  on 
meet  with  in  certain  electric  and  magnetic  combinations. 

26.  I.  An  infinitely  thin  homogeneous  spherical  shell 


24  INTRODUCTION. 

exercises  no  action  upon  a  mass  within  it— A  mass 
equal  to  unity  being  concentrated  at  P,  draw  from  this  point 
as  apex  an  elementary  cone  cutting  two  surfaces  ds  and  ds' 
on  the  sphere  at  distances  /and  /'.  Let  <r  be  the  surface 
density,  or  quantity  of  the  agent  per  unit  of  surface.  The 
elements  ds  and  ds'  are  consequently  charged  with  masses 

dm  =  ads, 

dm'  =  ads'. 

Their  actions  on  the  unit  mass  at  P  are 
kdm  _  kads 

~r~      r  ' 

kdm'      kcrds' 


r        r 

But  the  elements  ds  and  ds',  making  equal  angles  with  the 
axis  of  the  cone,  are  to  each  other  as  their  projections  per- 
pendicular to  this  axis,  and  as  these  latter  are  themselves 
proportional  to  the  square  of  their  distances  from  the  apex 
of  the  cone,  we  have 

ds_  _d/. 

r  :Z7*; 

whence 

kdm  _  kdmf 

~T~     ~r~~° 

As  the  whole  surface  of  the  sphere  can  be  divided  into 
pairs  of  elements  like  ds  and  d/,  whose  actions  neutralize 
each  other,  the  total  effect  of  the  shell  on  the  point  P,  or  on 
any  other  internal  point,  is  zero. 

We  conclude  from  this  that  the  potential  is  constant  in  all 
points  inside  of  a  spherical  shell. 

This  potential  is  therefore  the  same  as  that  of  the  centre, 
which  is  expressed  by 


A  P  PLICA  T10NS.  2  5 

Corollaries. — (a)  The  surface  of  the  shell  under  consider- 
ation  is  equipotential. 

(b)  The  potential  energy  of  the  shell  is  —k—-. 

2  A/ 


(c)  This  conclusion  would  <ftill  hold  in  the  case  of  a  series 
of  concentric  shells  acting  upon  an  internal  point,  which 
may,  at  the  limit,  be  situated  on  the  internal  surface  of  the 
innermost  shell. 

27.  II.  The  action  of  a  homogeneous  spherical  shell 
on  an  external  point  is  the  same  as  if  the  whole  mass 


FIG.  6. 

were  concentrated  at  the  centre  of  the  sphere.  —  If  unit 
mass  be  concentrated  at  Py  it  is  evident  that  by  symmetry 
the  resultant  action  must  be  directed  along  OP.  An  ele- 
ment ds  at  A  exerts  on  unit  mass  a  force  whose  component 
along  OP  is 

ka-^  cos  a. 


Pr  being  the  conjugate  point  of  P,  such  that  OP'  X  OP 
=  R\  connect  A   and  P'.     The  triangles  OAP'  and   OAP 


26  IN  TROD  UCTION. 

are  similar,  for  they  have  the  angle  A  OP  in  common  and 
the  adjacent  sides  proportional  ;  whence 


__    _ 
AP'  ":    R 

and,  substituting, 


But 

ds  cos  a 


o>  being  the  solid  angle  subtended  by  the  element  ds  at  the 
point  P'\  therefore 

r> 

dH  =  kcrdoo  X  -  • 
OP* 

The  action  of  the  whole  shell  will  be 


This  action  is  the  same  as  if  the  entire  mass  were  concen- 
trated in  the  point  O. 

Corollary.  —  For  a  point  infinitely  near  to  the  surface  the 
action  of  the  shell  would  be 


28.  Action  of  a  Homogeneous  Sphere  upon  an  Exter- 
nal Point.  —  If  the  sphere  were  composed  of  a  number  of 
similar  shells  superposed,  this  conclusion  would  still  hold. 
We  can  therefore  say  that  a  homogeneous  sphere,  or  sphere 
composed  of  homogeneous  shells,  acts  upon  an  external  point  as 
if  the  mass  were  concentrated  at  the  centre  of  the  sphere. 

Calling  in  this  case  #  the  mass  per  unit  volume,  we 
have 


OP 


APPLICA  TIONS.  27 

This  property  justifies  the  hypothesis  of  the  concentration 
in  physical  points  of  masses  which  in  reality  occupy  definite 
volumes  around  these  points. 

As  a  particular  case,  if  th^e  point  is  at  the  surface  of  the 
sphere,  the  preceding  expression  reduces  to 


ff=±> 

3 


29.  Action  of  a  Homogeneous  Sphere  upon  an  Inter- 
nal  Point. — If   the  point   were   inside   of  a   homogeneous 
sphere,  this  latter  could  be  divided  into  two  parts,  separated 
by  a  concentric  sphere  passing  through  the  given  point. 

The  action  of  the  external  portion  is  null;  the  action  of 
the  sphere  internal  to  the  point  is  equal  to  that  of  an  equal 
mass  concentrated  at  the  centre.  Denoting  by  /  the  dis- 
tance of  the  point  P  from  the  centre, 

H  =  *7tkld. 
3 

30.  Surface  Pressure. — In  the  case  of  a  homogeneous 


FIG.  7. 

spherical  shell  the  component  due  to  the  element  ds  de- 
pends only  on  the  solid  angle  subtended  by  it  at  the  point 


28  INTRODUCTION. 

P  '.     It  is  therefore  equal  to  that  of  the  element  ds',  corre- 
sponding to  ds. 

The  same  holds  for  all  the  elements  of  the  segment  adb, 
taken  in  pairs  with  those  of  the  segment  acb.  The  plane 
projected  on  ab  divides  the  sphere  into  two  zones  exercising 
the  same  actions  upon  P,  equal  to 


If  the  point  Pis  removed  indefinitely  from  the  sphere,  the 
two  segments  tend  to  become  equal.  If,  on  the  other  hand, 
the  point  P  approaches  indefinitely  to  the  spherical  shell, 
one  of  the  segments  has  as  its  limit  the  entire  sphere,  while 
the  other  tends  towards  zero. 

In  this  last  case  the  preceding  expression  becomes  2ttk(j. 

Now  we  have  just  seen  that  the  whole  shell  exerts  upon 
unit  mass,  situated  infinitely  near  to  its  surface,  an  action 
equal  to  ^nk<J.  It  follows  from  this  that  the  infinitely 
small  element  adjoining  the  point  exercises  an  action  equal 
to  that  of  the  entire  sphere  ;  and  in  fact  when  the  point  P 
traverses  the  shell  the  force  acting  upon  it  becomes  zero  ; 
during  this  infinitesimal  displacement  the  action  of  the 
spherical  shell  has  remained  constant,  but  that  of  the  sur- 
face element  next  to  the  point  has  changed  sign,  so  that  the 
resultant  is  zero. 

If  the  action  of  the  spherical  shell  on  unit  mass  situated 
at  its  surface  is  2nk<T,  it  will  be  2nkcr*  on  the  mass  <r  which 
charges  unit  surface. 

This  force,  with  which  the  shell  acts  upon  the  charge  of 
unit  surface,  is  called  surface  pressure. 

The  intensity  of  the  field  infinitely  near  the  surface  being 
H=  47T/£cr,  the  surface  tension  is  expressed  indifferently  by 

f-f  * 
2nkcr  ,    or  by    __ 

*      Ink 


AP PLICA  TIONS.  29 

31.  Potential  Due  to  an  Infinitely  Thin  Disk  Uniformly 
Charged. — Let  cr  be  the  surface  ^density  of  a  disk  projected 

<^H  «,«-«•£ 

A 

z 


FIG.  8. 

on  AB  (Fig.  8).     On  a  ring  concentric  with  the  disk,  with 
radius  r  and  thickness  dr,  the  charge  is  27rrdrcr. 

The  potential  due  to  this  ring  at  a  point  O  on  the  axis  of 
the  disk  and  at  a  distance  a  is 

dC7  = 

*V+*' 

The  potential  due  to  the  whole  disk,  whose  radius  is  R, 
will  be 

_     CR  27ikrdro- 
~Jo    V72~+^ 

The  intensity  of  the  field  at  the  point  O  is 

du  _    ,__,         ~  ,t        _*._,_  __CQS 


OL  being  the  plane  angle  subtended  by  the  radius  of  the  disk 
at  the  given  point. 

At  a  point  infinitely  near  the  surface  of  the  disk  we  have 

H  —  27tk(T. 

This  expression  represents  the  force  due  to  the  charge 
of  the  element  infinitely  near  to  the  point  (§  30).  The 
other  elements  exert  forces  which  mutually  neutralize  each 
other. 

It  will  be  noticed  that  2n(\  —  cos  or)  expresses  the  angle 
subtended  by  the  disk  at  the  point  O.  We  obtain  directly 
the  expression  for  the  intensity  of  the  field  by  considering 


30  1NTROD  UCTION. 

the  action  of  an  element  ds  of  the  disk  upon  unit  mass. 
Calling  a  the  angle  of  the  axis  with  the  right  line  which 
joins  the  element  to  the  point  O,  the  projection  of  the  force 
due  to  this  element  along  the  direction  of  the  axis  is 

dH  =  kads—  cos  a  =  kadGO, 

doo  expressing  the  solid  angle  subtended  by  the  element  at 
the  point  O. 

The  action  of  the  whole  disk  is  H  =  ka-a). 

When  the  unit  mass  is  infinitely  near  the  disk,  the  force 
is  2nk(T.  Moreover,  it  is  the  same  at  all  points  of  the  disk. 


MAGNETISM. 
» 

PROPERTIES   0F   MAGNETS. 

32.  Definitions. — The    name    magnet   is   given   to   those 
bodies  which  possess  the  property  of   attracting  iron  filings. 
The   lodestbne   or  magnetic   oxide   of   iron   possesses   this 
property  by  nature,  but  it  is  artificially  acquired  in  a  much 
higher  degree  by  iron  and  its  derivatives,  steel  and  cast  iron. 
Tempered  steel  is  the  substance  which   retains  in  the  great- 
est degree  the  attractive  power  developed  by  magnetization. 

When  a  magnetized  steel  bar  is  thrust  into  iron  filings,  it 
is  noticed  that  the  filings  cling  by  preference  to  certain 
parts  of  the  bar,  designated  by  the  name  of  poles.  These 
bars  generally  present  two  poles  separated  by  a  neutral 
region  having  a  feeble  or  no  action  upon  iron  filings. 

33.  Action  of  the  Earth  on  a  Magnet. — When  a  magnet 
is  suspended  by  its  centre   of   gravity,  one  of  its  poles  is  in- 
variably directed  towards  the  north,  the  other  towards  the 
south.     For  this  reason  the  first  pole  is  called  the  north  or 
N-/0/^,  th2  second,  the  south  or  $-pole. 

34.  Law  of  Magnetic  Attractions. — If  several  magnets 
are  placed  near  each  other,  it  is  observed  that  the  poles  of 
the  same  name  repel  each  other  and  that  those  of  contrary 
name  attract  each  other.     The  study  of  these  actions  pre- 
sents some  difficulties,  because  it  is  not   possible  to  investi- 
gate the  reciprocal  action  of  two  isolated  poles. 

Coulomb,  however,  observed  that  long  magnetized  rods 
have  their  centres  of  action  near  their  extremities.  By 
bringing  the  poles  of  two  of  these  rods  sufficiently  near 
together  he  was  able  to  almost  entirely  eliminate  the  action 

31 


3  2  MAGNETISM. 

of  the  opposite  poles.  He  discovered  experimentally  that 
magnetic  forces  decrease  in  inverse  ratio  to  the  square  of  the 
distance  between  the  acting  pole*. 

We  shall  see  further  on  that  this  law  has  been  rigorously 
verified  by  Gauss. 

Magnetic  actions  are  therefore  to  be  classed  among  the 
Newtonian  forces,  and  we  may  apply  to  them  the  general 
theorems  demonstrated  in  the  introduction  to  this  work, 
the  agent  acting  in1  this  case  being  magnetism. 

We  shall  designate  by  quantity  of  magnetism  or  mass  of  a 
pole  a  quantity  proportional  to  the  force  which  it  exerts 
upon  neighboring  poles. 

Let  m  and  m'  be  the  quantities  of  magnetism  of  two 
poles:  their  mutual  reaction  is  by  definition  proportional  to 
m  and  m'  ,  and  consequently  to  the  product  mm'. 

The  expression  of  the  force  is  therefore 


/  expressing  the  distance  between  the  two  poles. 

35.  Unit  Pole.  —  The  coefficient  k  in  the  preceding  ex- 
pression may  be  considered  arbitrary  and  taken  equal  to 
unity.  In  reality,  the  coefficient  k  varies  with  the  medium 
in  which  the  magnetic  bodies  are  situated,  but  the  differ- 
ences in  the  various  gaseous  media  in  this  respect  are 
very  small  at  ordinary  temperatures.  Consequently  the  unit 
quantity  of  magnetism  is  the  quantity  which  repels  an  equal 
quantity,  at  a  distance  of  one  centimetre,  with  the  force  of 
one  dyne;  its  dimensions  are 


If  the  poles  are  of  contrary  name,  the  force  becomes 
attractive.  To  deduce  this  from  the  preceding  formula,  we 
need  only  give  opposite  signs  to  poles  of  contrary  name.  It 


PR  OPER  TIES   OF  MA  GNE  TS,  3  3 

has  been  settled  that  the  N  pole  shall  have  the  -f  sign,  and 
the  5  pole  the  —  sign. 

36.  Definitions.  —  Applying  the  general  definitions 
adopted  in  the  Introduction,'  we  will  call  the  space  in  which 
magnetic  forces  exist"  a  magnetic  field.  The  intensity  at  a 
point  of  the  field  is  measured  by  the  action  it  exerted  there 
on  positive  unit  mass.  The  direction  of  this  action  gives 
the  direction  of  the  field.  A  magnetic  line  of  force  repre- 
sents the  path  of  an  infinitesimal  positive  mass  free  to  move 
in  the  field.  The  field  possesses  a  potential  called  magnetic 
potential,  whose  expression  for  a  point  situated  at  distances 
/,  /',  I"  .  .  .  from  masses  m,  m',  m"  ...  is 


each  mass  being  given  its  own  sign. 

By  this  definition  the  dimensions  of  magnetic  potential 


are 


The  component  of  field  intensity  in  a  direction  /,  at  a 
point  where  the  potential  is  U,  is  expressed  by 


*=- 


dimensions, 


This  expression  represents,  as  we  have  seen  in  §23,  the 
flux  of  force  per  unit  of  surface  normal  to  d/. 

The  dimensions  of  field  intensity  differ  from  those  of  a 
force.  It  is  the  force  per  unit  pole,  and  its  dimensions  are 
those  of  a  force  divided  by  those  of  a  magnetic  mass. 

The  most  intense  fields  that  have  been  produced  up  to 
the  present  reach  about  30,000  C.  G.  S.  units,  or  30  kilo- 


34  MAGNETISM. 

gausses  ;  such  a  field  develops  a  force  of  30,000  dynes,  or 
about  30  grams,  upon  unit  pole. 

37.  Action  of  a  Uniform  Field  on  a  Magnet.  —  Consider 
a  uniform  field,  in  which  the  intensity  3C  is  constant  in 
magnitude  and  direction.  Experiment  shows  that  in  such 
a  field  a  magnet  is  not  subjected  to  any  force  of  translation, 
but  that  it  simply  tends  to  place  itself  in  a  definite  direc- 
tion. Hence  we  conclude  that  the  sum  of  the  positive 
masses  of  the  bar  is  equal  to  the  sum  of  the  negative 
masses,  since  the  resultant  of  the  actions  ot  the  field  on  the 
first,  or  3C.2m,  is  balanced  by  the  resultant  3C^(—  m)  of  the 
actions  on  the  second. 

These  resultants  are  applied  at  two  points  which,  in  the 
mathematical  theory  of  magnetism,  receive  more  particularly 
the  name  of  poles.  It  must  be  observed,  however,  that  the 
poles  which  are  thus  defined  have  no  more  physical  exist- 
ence than  has  the  centre  of  gravity  of  a  body.  An  imagi- 
nary line  passing  through  the  poles  is  called  the  magnetic 
axis  of  the  magnet.  The  distance  /  between  the  poles  is 
the  true  length  of  the  magnet. 

The  product  of  the  magnetic  mass  at  one  pole  by  the  dis- 
tance between  the  poles  is  the  magnetic  moment  of  the  bar 


12  m  —  9TI. 

Designating  by  ft  the  angle  of  the  magnetic  axis  with  the 
direction  of  the  field,  the  couple  acting  on  the  magnet  is  ex- 
pressed by 

OC/sin  fi2m  =  3C371  sin  ft. 

If  the  magnet  be  suspended  by  its  centre  of  gravity,  the 
duration  of  a  complete  oscillation  of  small  amplitude  is 


/AT1 

= 2  V  ^' 


PROPERTIES   OF  MAGNETS.  35 

where  K*  is  the  moment  of  inertia  of  the  magnet,  and  w 
the  maximum  couple,  equal  to 


38.  Terrestrial  Magnetic  -Field.  —  Experiment  shows 
that  the  duration  of  the  oscillation  of  a  magnet  is  constant 
within  the  limits  of  a'laboratory  room,  provided  that  there 
be  no  other  magnetic  mass  in  the  room  or  near  by.  It  may 
therefore  be  assumed  that,  in  a  space  of  small  extent,  the 
earth  develops  a  uniform  field.  The  vertical  plane  passing 
through  the  axis  of  a  magnet  hanging  freely  is  called  the 
magnetic  meridian  of  a  place. 

The  declination  is  the  angle  of  this  plane  with  the  geogra- 
phical meridian  ;  the  inclination,  the  angle  of  the  axis  of  the 
magnet  with  the  horizontal.  In  our  hemisphere  the  north 
pole  of  magnets  dips  below  the  horizon,  so  that  a  weight 
must  be  placed  on  the  south  pole  to  make  the  oscillations 
take  place  in  the  horizontal  plane.  The  component  of  the 
intensity  of  the  terrestrial  field  in  this  plane  is  called  the 
horizontal  component. 

The  terrestrial  lines  of  force,  which,  in  a  limited  space, 
may  be  considered  as  parallel,  really  converge  towards 
points  called  magnetic  poles,  which  oscillate  in  the  neighbor- 
hood of  the  geographical  poles. 

The  following  numerical  data,  due  to  Airy,  show  the 
mean  magnetic  values  for  Greenwich,  t  being  the  date  : 

Declination  :   19°  12.1'  —  (t  —  1876)  X  7.38'. 

Horizontal  component:  0.1797  -\-  (t  —  1876)  X  0.00027. 

Inclination:  67°  40.3'  —  (t  —  1876)  X  2.04.'* 

We  see  from  this  that  the  terrestrial  magnetic  field  has 
only  a  feeble  intensity.  It  will  be  shown  later  on  that  it  is 
possible  to  produce  very  much  intenser  fields  in  the  interior 


*  For  the  study  of  terrestrial  magnetism  consult  Gauss,  Allgemeine  Theorie 
dfs  Erdmagneiismus;  Mascart  et  Joubert,  Lemons  sur  V Electricity  ft  sur  If 
Magnttisnif,Vo\.  I. 


36  MAGNETISM. 

of  a  bobbin  of  wire  traversed  by  an  electric  current.  In 
such  a  bobbin,  if  the  length  is  very  great  in  proportion  to  the 
cross-section,  the  field  may  be  considered  uniform. 

39.  Weber's  Hypothesis. — When  a  magnet  is  broken 
across  its  neutral  region,  we  do  not  get  two  isolated  poles, 
but  two  new  magnets.  The  original  magnet  can  be  restored 
by  joining  the  broken  pieces  together  again ;  the  poles 
which  were  developed  at  the  surfaces  of  rupture  neutralize 
each  other.  This  fact  of  the  division  of  a  magnet  always 
urnishing  complete  magnets,  however  small  the  fragments, 
may  be,  has  led  Weber  to  suppose  that  the  polarization 
takes  places  in  the  molecules  composing  the  bar,  each  one 
of  them  being  a  complete  magnet  possessing  two  poles.* 
The  neutral  state,  by  this  hypothesis,  results  from  the  non- 
orientation  of  the  molecules,  whose  poles  mutually  neutral- 
ize each  other.  But  if  a  neutral  bar  is  placed  in  a  field,  the 
magnetic  axes  of  the  molecules  are  drawn  into  their  proper 
positions:  the  N-poles  in  the  direction  of  the  field,  and  the 
S-poles  the  opposite  way.  To  explain  the  observed  varia- 
tions in  the  degree  of  magnetization,  we  have  to  assume  that 
the  molecules  oppose  a  certain  resistance  to  this  alignment, 
varying  according  to  the  physical  condition  of  the  bar,  and 
to  which  the  name  coercive  force  has  been  given. 

This  resistance,  an  explanation  of  which  by  Ewing  will  be 
seen  latter  on,  is  feeble  in  annealed  or  soft  iron,  so  that 
when  a  bar  of  this  metal  is  introduced  into  a  magnetic  field 
of  medium  intensity  it  becomes  strongly  magnetized.  But 
it  readily  loses  its  magnetization  when  removed  from  the 
field  and  retains  only  traces  of  residual  magnetism.  The 
name  magnetizing  force  is  given  to  the  intensity  of  the  field 
which  induces  the  magnetization. 

*  See  Maxwell,  Electricity  and  Magnetism,  Vol.  II.- 


PROPERTIES   OF   MAGNETS.  37 

Cold  hammering  increases  the  coercive  force  of  iron,  but 
this  force  is  especially  increased  by  combining  the  metal 
with  certain  foreign  substances,  such  as  carbon,  tungsten 
and  chromium,  in  small  proportions.  A  steel  bar  attains  its 
maximum  coercive  force  when4t  has  been  heated  to  a  bright 
red,  and  tempered  either  by  sudden  chilling  in  oil,  water,  or 
mercury,  or  by  powerful  pressure  under  an  hydraulic  press 
during  its  cooling.  This  last  method  gives  the  metal  a  more 
uniform  hardness  than  tempering  by  immersion.  The  degree 
of  annealing  may  be  estimated  by  the  electric  conductiv- 
ity of  samples.  This  conductivity  increases  from  its  original 
value  for  soft  steel  to  three  times  that  value  for  steel  tem- 
pered in  mercury.  A  tempered  steel  bar  is  more  difficult  to 
magnetize  than  iron,  but  it  retains  a  considerable  perma- 
nent magnetization. 

Different  facts  tend  to  corroborate  Weber's  hypothesis. 

1.  The  magnetization  in  a  field  which  is  increasing  in  in- 
tensity tends  towards  a  limit  called  saturation,  which  prob- 
ably corresponds  to  the   parallelism  of  the   molecular  axes 
with  the  direction  of  the  field. 

2.  Every  cause  of  molecular  disturbance  favors  the  mag- 
netization of  a  bar  subjected   to  a  magnetizing  force,  and 
also  favors  its  demagnetization  after   it  has  been  withdrawn 
from  the  field.     Thus  a  bar  of  soft  iron  placed  vertically  is 
magnetized  by  the  action  of  the  vertical  component  of  ter- 
restrial magnetism  when  it  is  struck  lightly. 

The  bar  retains  its  magnetization  when  placed  in  a  differ- 
ent position,  provided  it  be  kept  free  from  vibrations,  but  a 
slight  blow  is  sufficient  to  dissipate  its  magnetism.  Vibra- 
tions have  a  specially  marked  effect  upon  iron.  Ewing  has 
shown  that  if  a  bar  of  this  metal  be  kept  from  the  slightest 
vibration  one  can  obtain  residual  magnetizations  much 
greater  than  those  shown  in  steel  bars  ;  but  the  least  vibra- 
tion causes  the  acquired  magnetism  to  vanish  almost  com- 


38  MAGNETISM. 

pletely.  A  violent  blow  can  likewise  take  from  a  recently 
magnetized  bar  of  steel  more  than  half  its  magnetic  moment, 
but  successive  blows  produce  a  more  and  more  feeble  effect. 

The  same  holds  for  variations  of  temperature.  Raising 
the  temperature  weakens  the  power  of  a  magnet.  At  a 
bright  red  heat  the  magnetization  disappears  entirely. 

If  a  magnetized  and  tempered  bar  is  annealed  at  a  tem- 
perature of  100°,  for  example,  constancy  of  the  magnet- 
ization for  lower  temperatures  is  secured  ;  that  is  to  say, 
variations  less  than  100°  cause  only  a  temporary  change  in 
the  magnetic  moment  of  the  bar,  which  resumes  the  same 
value  at  the  same  temperature. 

The  variations  of  the  magnetic  moment  are  then  sensibly 
a  linear  function  of  the  temperature  ;  let  3T10  be  the  moment 
at  o°  C.,  m,t  the  moment  at  8°  C. : 

SHI,  =  Wl0(i  —aff). 

3.  Magnetization  produces  a  slight  progressive  elongation 
of  the  magnetized  bars  up  to  a  certain  limit,  beyond  which 
a  contraction  takes  place,  as  if  the  molecules,  after  being 
oriented,  tended  to  approach  each  other  and  to  diminish  the 
intermolecular  space  (Bidwell). 

When  a  bar  is  subjected  to  variable  magnetizing  forces,  it 
produces  a  sound,  which  may  be  attributed  to  the  displace- 
ments of  air-molecules  caused  by  the  dilatations  or  contrac- 
tions of  the  magnet. 

4.  Beetz  has  shown  that  a  feeble  magnetizing  force  applied 
to  iron  at  the   moment  of   its   precipitation   by  electrolysis 
magnetizes  it  to  saturation,  which  is  the  result  of  the  fact 
that  the  molecules  of  the  metal  change  their  position  freely 
at  the  moment  of  reduction. 

40.  Elementary  Magnets.  Intensity  of  Magnetization. 
— The  hypothesis  just  developed  leads  us  to  analyze  the 
properties  of  elementary  magnets,  whose  length  is  taken  as 


PROPERTIES   OF  MAGNETS.  39 

infinitely  small  in  comparison  with  the  finite  distances  of  the 
field. 

Intensity  of  magnetization  of  an  elementary  magnet  is  the 
name  given  to  the  ratio  of  its  magnetic  moment  to  its 
volume. 

0=       ;     dimensions,    \L~^M^T~l\ 

V 

Elementary  magnets  are,  by  assumption,  cylindrical  in 
form,  with  their  poles  concentrated  on  the  two  ends.  Under 
these  conditions  we  call  the  ratio  of  the  magnetic  mass  of 
the  poles  to  their  surface  the  density  of  the  poles : 

or  =  . 

s 

Now,  calling  the  length  of  the  magnet  /,  we  have  ml  =  2fll, 
si  —  V\  it  follows  that  the  density  and  intensity  of  magnet- 
ization have  the  same  numerical  expression  and  the  same 
dimensions. 

Let  SN  (Fig.  9)  be  an  elementary  magnet  whose  poles  of 

•P 

V~~" 

FIG.  9. 

mass  m  are  at  distances  Z,  L'  from  a  point  P.    The  magnetic 
potential  in  this  point  is 


L  \L'     Li  LL' 

As  L  differs  from  L'  by  only  an  infinitely  small  quantity, 
we  can  replace  LL'  by  D,  and  L  —  L  by  /  cos  /?,  j3  being  the 
angle  made  by  the  axis  of  the  magnet  SN  with  the  right  line 
drawn  from  5  to  P.  Consequently 

7-7       ml  cos  0      3ft  cos  8 

(J    =:     _ zz:    

D  D 


40  MAGNETISM. 

41.  Magnetic  or  Solenoidal  Filament. — If  we  place  ele- 
mentary magnets  of  equal  intensity  of  magnetization  end  to 
end,  the  adjoining  poles  neutralize  each  other,  and  there  re- 
main only  two  resultant  poles  at  the  ends  of  the  chain, 
which  is  called  a  magnetic  filament  or  solenoidal  filament. 


FIG.  10. 


The  potential  at  a  point  P,  situated  at  distances  L  and  U 
from  the  resultant  poles,  is 


which  expression  is  independent  of  the  form  of  the  mag- 
netic filament,  but  depends  solely  on  the  position  of  its 
extremities.  Consequently  a  magnetic  filament  forming  a 
closed  chain  has  a  zero  potential  for  all  external  points;  that 


PROPERTIES   OF  MAGNETS.  41 

is,  a  closed  magnetic   filament    exercises   no    action    upon 
neighboring  magnetic  masses. 
The  equation 

m(ji  —  j\  —  constant 

represents  an  equipotential  surface. 

Adopting  the  graphic  method  shown  in  §  16,  we  can  draw 
the  equipotential  lines  in  a  plane  passing  through  the 
masses  -f-  m  and  —  m. 

Fig.  10  shows  the  equipotential  lines  due  to  two  such 
masses  ;  they  are  ovoid  curves  enveloping  the  poles.  The 
vertical  axis  of  symmetry  is  equipotential.  The  lines  of 
force,  whose  direction  is  shown  by  the  arrows,  cut  the  equi- 
potential lines  orthogonally. 

42.  Uniform  Magnets. — Imagine  a  right  circular  cylin- 
der formed  of  equal  adjoining  rectilinear  filaments.  The 
effect  of  such  a  combination,  as  regards  external  points, 
is  the  same  as  that  of  two  magnetic  shells,  of  equal  and  op- 
posite densities,  situated  at  the  extremities  of  the  cylinder. 
The  intensity  of  magnetization  is  constant  at  every  point  of 
the  cylinder. 

The  action  of  a  right  cylinder,  uniformly  magnetized,  at 
an  exterior  point  on  the  prolongation  of  its  axis  is,  denoting 
by  a  and  a'  the  plane  angles  subtended  by  the  radii  of  the 
bases  of  the  cylinder  at  the  given  point  (§  31), 
3C  =  27r3(cos  a'  —  cos  a). 

We  can  likewise  place  together  rectilinear  filaments  of 
equal  intensity  of  magnetization,  but  of  different  lengths, 
whose  extremities  abut  on  the  surface  of  a  sphere.  A  spher- 
ical magnet  thus  defined  presents  two  hemispherical  shells 
of  opposite  signs. 

The  density  of  these  shells  is  equal  to  the  intensity  of  the 
filaments  at  the  extremities  of  the  diameter  NS,  where  the 


42  MAGNETISM. 

elements  cut  the  surface  of  the  sphere  normally.  At  other 
points  the  density  decreases  as  the  cosine  of  the  angle  <*,  a 
representing  the  inclination  of  the  filaments  to  the  radii  of 
the  sphere.  It  is  zero  along  the  great  circle  normal  to  NS. 

Such  a  distribution  may  be  represented  by  slipping  a 
positive  sphere,  with  centre  O,  over  an  equal  sphere,  but  of 
contrary  sign,  with  centre  at  O',  the  distance  between  the 
centres  being  infinitesimal,  so  that  OO'  represents  the  maxi- 
mum thickness  of  the  magnetic  shell. 

In  the  meridian  section  represented  in  Fig.  n,  the  shells 
bounded  by  the  surfaces  of  separation  of  the  two  spheres 


FIG.  ii. 

have  a  thickness  decreasing  as  the  cosine  of  the  angle  a. 
The  region  common  to  the  two  is  evidently  neutral. 

The  effect  of  the  system  on  an  internal  point  P  is  equal 
to  the  resultant  of  the  actions  of  both  spheres. 

Now  calling  d  the  cubic  density  of  the  masses,  the  effect 
of  the  sphere  O  on  unit  pole  situated  at  P  is  (§  29) 

Ijrtf  X  OP. 

3 

This  action,  directed  along  OP,  may  be  represented  by 
the  length  of  this  line.  The  negative  sphere  exercises  an 
action  equal  to 

-±ndX  O'P, 
3 

and  represented  by  PO'. 


PROPERTIES   OF  MAGNETS.  43 

The  resultant  of  these  two  forces  is 

'*-nd  X  OO', 
3 

since  OO '  closes  the  triangle  formed  by  the  components  OP 
and  PO'.  Now  d  X  OO'  —  cr Represents  the  maximum  mag- 
netic density  of  the  shell. 

The  resultant  of  the  actions  of  a  uniformly  magnetized 
sphere  is  therefore  constant  in  magnitude  and  direction  for  all 

internal  points,  and  equal  to  ^n  multiplied  by  the  maximum 

density  or  intensity  of  magnetization  of  the  sphere. 

43.  Magnetic  Shells. — This  name  is  given  to  a  lamellar 
magnet  whose  faces  have  equal  and  opposite  densities.  A 
magnetic  shell  may  be  considered  as  the  result  of  the  juxta- 
position of  elementary  magnets. 

Let  cr  be  the  density  of  the  faces,  equal  to  the  intensity 
of  magnetization  of  the  component  elements ;  e  the  thick- 
ness of  the  shell,  that  is,  the  length  of  the  elements. 


FIG.  12. 

The  product  ecr  =  SF,,  which  represents  the  moment  of 
the  shell  per  unit  surface,  is  called  the  strength  of  the  shell. 
The  dimensions  of  this  quantity,  the  importance  of  which 
will  be  understood  in  the  study  of  electromagnetism,  are 


Let  C  (Fig.  12)  be  a  curved  shell  with  its  positive  face 


44  MAGNETISM. 

turned  towards  the  interior.  The  potential  at  a  point  O  is 
the  sum  of  the  potentials  due  to  the  component  elementary 
magnets.  For  an  element  dj,  whose  axis  makes  an  angle  ft 
with  the  right  line  joining  it  to  the  point  O,  situated  at  a 
distance  L,  the  potential  is 

ATT  —  e°"dj  cos  ft 

-~L— 

but     S  C°S  -  represents  the  solid   angle  subtended  by  the 
J^i 

surface  ds  at  the  point  O.     In  order  to  make  this  notation 
uniform  with  that  which  will  be  met  with  in  electromagnet- 
ism,  we  shall  consider  the  solid  angle  as  positive  when  the 
point  O  faces  the  S  pole  of  the  magnetic  element. 
We  have  then 


Extending  the  integration  to  all  the  elements  of  the  shell, 
and  observing  that  all  the  elementary  cones  which  cut  the 


FIG.  13. 

shell  twice  give  equal  and  contrary  elements  of  potential, 
we  get  for  the  total  potential 

U=-  5. ft. 

The  potential  due  to  a  shell  at  any  point  is  equal  to  the 
product  of  the  strength  of  the  shell  by  the  solid  angle  subtended 
at  that  point  by  the  contour  of  the  shell. 

44.  Corollary. — If  the  point  O  were  at  the  interior  of  the 
shell,  the  solid  angle  would  be  -|-  (4?r  —  ft').  Consequently, 


PROPERTIES   OF  MAGNETS.  45 

if  the  point  passes  from  a  point  O  to  a  point  O'  infinitely 
near  on  the  other  side  of  the  shell,  the  potential  varies  from 
-f  SF,(47r  —  /?')  to  —  SF,/ff',  that  is,  by  the  quantity  472:  9> 

The  work  done  by  unit  ppsitive  mass,  in  passing  from  a 
point  on  the  surface  of  a  shgll  to  a  point  infinitely  near 
situated  on  the  other^side,  is  equal  to  4?r  multiplied  by  the 
strength  of  the  shell.  This  work  is,  moreover,  independent 
of  the  path  followed  between  these  two  points  (§  12). 

45.  Energy  of  a  Shell  in  a  Field.  —  Let  us  consider  a 
field  of  force  due  to  a  pole  m  situated  at  a  point  O  (Fig.  12)\ 
the  work  expended  to  bring  the  shell  to  its  present  position 
represents  the  relative  energy  of  the  shell  and  the  field.  It 
is  equal  to  the  work  required  to  bring  the  mass  m  to  the 
point  O  whose  potential  is  U,  or 


Now  m/3  is  the  flux  of  force  from  the  pole  across  the  solid 
angle  limited  by  the  contour  of  the  shell. 

We  will  call  this  flux  $,  considering  it  as  positive  when  it 
enters  by  the  negative  face  of  the  shell,  and  as  negative 
when  it  enters  by  the  positive  face: 

W=  -  JF,$. 

If  the  field  is  produced  by  several  poles  m,  m',  mn  ',  the 
total  energy  will  become 


The  relative  energy  of  a  shell  and  a  field  is  therefore  equal 
to  the  product  of  the  strength  of  the  shell  by  the  flux  included 
within  the  contour  of  the  shell. 

If  the  flux  penetrates,  as  in  Fig.  12,  by  the  positive  face, 
$  is  negative,  and  the  product  takes  the  plus  sign. 

When  a  shell  is  free  to  move  in  afield,  it  tends  to  move  so  that 
the  expression  for  the  potential  energy  becomes  a  minimum  ; 


46  MAGNETISM. 

that  is,  the  flux  entering  by  the  negative  face  tends  towards 
a  maximum.  It  will  easily  be  shown  that  this  condition  is 
satisfied  in  the  case  of  a  shell  and  a  positive  pole  when  the 
latter  touches  the  negative  face.  A  plane  shell  situated  in 
a  uniform  field  will  take  up  a  position  normal  to  the  direc- 
tion of  the  field,  so  that  the  lines  of  force  penetrate  it  by 
the  S-face. 

46.  Relative  Energy  of  Two  Shells.  —  Let  us  consider 
two  neighboring  shells  A,  A',  of  strengths  SF,  and  IF/.  Let  $' 
be  the  flux  of  force  from  ^['across  the  section  of  A  entering 
by  its  negative  face.  The  energy  of  the  shell  A  is,  as  we 
have  just  seen,  expressed  by 


Now  the  flux  $'  may  be  represented  by  the  product  of  SF' 
into  a  factor  Lm  ;  whence 

W=-SS'Lm  ........    (i) 

This  expression  must  evidently  represent  the  energy  of 
the  shell  A',  for  the  same  work  is  expended  to  bring  the 
shell  A'  up  to  A  as  to  bring  A  up  to  A  '  .  But  as  the 
energy  of  A  '  is  also  given  by  the  product  of  SF/  into  th-e 
flux  #  passing  from  A  to  A',  we  see  that  $  =  &,Lm,  just 
as  we  found  $'  =  &s'Lm\  whence  we  obtain 


The  factor  Lmj  called  coefficient  of  mutual  induction  of  the 
two  shells,  represents,  as  seen  above,  the  ratio  of  the  flux 
across  one  of  the  shells  to  the  strength  of  the  neighboring 
shell.  Equation  (i)  shows  that  the  dimensions  of  Lm  reduce 
to  [L]. 

47.  Artificial  Magnets.—  Instead  of  presenting,  like  uni- 
form magnets,  a  surface  distribution  of  magnetism,  magnet- 


PROPERTIES   OF  MAGNETS. 


47 


ized  bars  possess  free  magnetic  masses  internally.  We  can 
even  superpose  opposite  magnetizations  in  a  steel  bar  by 
submitting  it  successively  to  magnetizing  forces  of  opposite 
directions.  When  such  a  bar  is  dissolved  in  an  acid,  there 
appear  progressively  layers  rrjagnetized  in  opposite  direc- 
tions. 

This  experiment  proves  that  the  magnetization  affects  at 
first  the  superficial  layers  of  the  bar,  which  are  moreover 
tempered  harder  than  the  inner  layers.  Hence  the  utility 
of  employing  thin  plates  of  steel,  separately  magnetized,  in 
order  to  obtain  powerful  magnets. 

We  can  show  in  a  striking  way  the  form  of  the  magnetic 


FIG.  14. 

field  due  to  a  bar  magnet  by  placing  over  it  a  sheet  of 
paper  covered  with  iron  filings.  The  particles  of  iron  be- 
come magnetized  by  induction,  orient  themselves  along  the 
directions  of  the  field,  and  arrange  themselves  in  continuous 
rows,  representing  the  lines  of  force. 

The  image  thus  produced  may  be  fixed  if  sensitized  pho- 
tographic paper  be  used  and  then  exposed  to  the  actinic  ac- 


48  MAGNETISM. 

tion  of  light  while  covered  with  the  filings.  The  develop- 
ment of  the  image  shows  the  shadows  produced  by  the 
filings.  Fig.  14  thus  represents  the  magnetic  field  of  two 
adjacent  bar  magnets. 

By  observing  the  distribution  of  iron  filings  in  the  field  of 
a  single  or  of  several  magnets,  and  the  curious  patterns  re- 
produced by  the  particles,  Faraday  was  led  to  the  idea  that 
the  seat  of  the  magnetic  forces  is  in  the  medium  which 
separates  the  acting  poles.  According  to  him,  lines  of  force 
are  not  a  mere  mathematical  conception,  but  have  a  real 
existence  corresponding  to  a  particular  state  of  the  space 
around  the  poles.  Faraday  imagined  this  medium  as  being 
strained  along  the  lines  of  force,  and  he  readily  substituted 
mentally  for  these  lines  elastic  threads  having  a  tendency  to 
contract  and  thus  cause  neighboring  poles  to  approach. 

To  explain  the  curvature  of  the  lines  of  force,  Faraday 
assumed  that  they  repel  each  other  when  they  proceed  in 
the  same  direction,  so  that  each  of  them  takes  a  curved 
form  whose  tendency  to  return  to  a  rectilinear  form 
balances  the  repulsion  of  the  neighboring  lines. 

Although  experiment  shows  that  a  magnetized  bar  has 
free  magnetic  masses  within  it,  it  is  possible  to  imagine  a 
surface  distribution  of  magnetism  producing  the  same  exter- 
nal field  as  the  actual  distribution.  Suppose  the  magnet, 
for  example,  to  be  formed  of  longitudinal  magnetic  filaments 
some  of  which  end  on  the  end  faces  of  the  magnet  and 
others  on  its  lateral  surfaces.  Then  the  poles  of  the  fila- 
ments constitute  the  surface  charges  of  the  magnet,  as  we 
have  seen  in  the  case  of  a  magnetized  sphere  (§  42),  and  the 
curves  taken  by  the  iron  filings  (Fig.  14),  may  be  considered 
as  the  prolongation  of  the  axes  of  the  filaments. 

To  determine  the  imaginary  distribution  of  magnetism 
giving  the  same  external  results  as  the  real  distribution,  we 
measure  the  variation  of  the  field  intensity  around  the  mag- 


PROPERTIES   OF  MAGNETS.  49 

net  along  its  axis.  To  do  this,  we  determine  the  period  of 
oscillation  of  a  small  magnetized  needle  moving  on  a  pivot, 
and  which  is  brought  successively  to  the  various  points 
where  it  is  desired  to  know  the  intensity.  This  means  is 
not,  however,  rigorous,  because  the  force  is  not  the  same  at 
both  poles  of  the  needle,  and  the  latter's  magnetism  may  be 
altered  under  the  influence  of  the  field  that  is  being  investi- 
gated. By  this  experiment  we  find  that  the  field  decreases 
rapidly  from  the  extremity  of  the  magnet  towards  the  mid- 
dle, unless  there  be  intermediate  or  consequent  poles.  In  long 
needles  of  hard  steel  the  poles  are  very  near  the  ends,  and 
the  neutral  line  extends  over  the  greatest  part  of  their 
length.  In  every  case  the  neighborhood  of  the  edges  gives  a 
more  intense  field  than  the  neighborhood  of  the  plane  surfaces. 


FIG.  15. 

Fig.  15  shows  the  curves  obtained  by  marking  off  on  the 
perpendiculars  to  the  axis  of  the  bar  lengths  proportional  to 
the  components  of  the  field  along  these  lines.  The  ordinates 
are  inversely  proportional  to  the  square  of  the  periods  of  os- 
cillation of  the  magnetized  needle  opposite  various  points  of 
the  axis  and  at  the  same  distance  from  it,  and  oscillating  in  a 
plane  normal  to  the  magnet.  These  ordinates  may  be  con- 
sidered as  proportional  to  the  thickness  of  the  magnetic 
shell  having  the  same  external  effect  as  the  real  distribution 
of  magnetism. 

Magnets  undergo  a  slow  demagnetization,  which  may  be 
explained  by  the  repulsion  exercised  between  poles  of  the 
same  name  in  neighboring  molecules.  This  loss  is  retarded 
by  joining  the  poles  by  a  piece  of  soft  iron  called  armature®? 
keeper.  Opposite  poles  are  developed  in  this  latter  which 


5O  MAGNETISM. 

retain  the  magnetization  of  the  magnet,  since  closed  mag- 
netic filaments  are  formed  through  the  armature,  and  the 
poles  of  the  elements  composing  these  filaments  attract  and 
neutralize  each  other  in  couples. 

The  best  steels  for  permanent  magnets  are  those  capable 
of  acquiring  the  hardest  temper. 

The  addition  of  3  per  cent,  of  tungsten  increases  the 
coercive  force  of  steel  very  perceptibly.  The  tempering 
may  be  done  in  oil,  water,  or  mercury.  The  bath  should  be 
of  sufficient  volume  to  prevent  great  rise  in  temperature 
and  splashing  of  the  liquid.  According  to  Strouhal  and 
Barus,  the  best  way  to  obtain  a  powerful  and  constant  mag- 
net is  to  make  the  steel  as  hard  as  possible  by  tempering, 
and  then  to  anneal  it  for  20  to  30  hours  in  steam  at  100°  C. 
It  is  next  magnetized  by  placing  its  extremities  on  the  poles 
of  a  powerful  electromagnet,  and  finally  annealing  it  again 
for  at  least  5  hours  in  steam.  This  method  secures  a 
magnetization  which  resists,  as  far  as  is  possible,  both  blows 
and  the  daily  variations  of  temperature. 

According  to  Preece,  the  intensity  of  permanent  magnet- 
ization obtainable  in  prisms  of  I  cm  section  and  10  cm 
length,  made  of  good  magnet  steel  bearing  the  stamp  Mar- 
chal,  Clemandot  &  Allevard,  varies  from  100  to  225  C.  G.  S. 
units.  These  numbers  express  the  ratio  of  the  permanent 
moment  of  the  magnets  to  their  volume. 

48.  Determination  of  the  Magnetic  Moment  of  a 
Magnet.  Magnetometer. — When  we  cause  a  magnet  to 
oscillate  horizontally  in  the  earth's  magnetic  field,  the  mag- 
net being  hung  by  a  thread  having  no  sensible  torsion-pull, 
the  duration  of  a  complete  oscillation,  of  sufficiently  small 
amplitude,  is 

/~H 

-2*V  gft  3C 


PROPERTIES   OF  MAGNETS.  5  I 

K  being  the  moment  of  inertia  of  the  magnet,  311  its  mag- 
netic moment,  and  5C  the  horizontal  component  of  the 
earth's  field. 

From  this  we  deduce 


(i) 


If  3C  be  known,  the  magnetic  moment  can  be  determined 
from  this  equation. 

If  this  is  not  the  case,  a  second  experiment  may  be  made, 
The  magnet  being  placed  in  a  horizontal  plane  and  normal 
to  the  magnetic  meridian,  a  small  magnetized  needle  is  hung 


FIG.  1 6. 

at  a  certain  distance  on  the  prolongation  of  the  given  mag- 
net's axis  (Fig.  16). 

Let  2/  be  the  distance  between  the  poles  of  the  magnet, 
2/'  that  between  the  poles  of  the  needle,  L  the  distance 
from  the  centre  of  the  magnet  to  that  of  the  needle,  -f-  m 
and  —  m  the  poles  of  the  magnet,  +  m'  and  —  m'  the  poles 
of  the  needle. 

The  magnetic  moments  are  respectively 

9fTl  =  2ml, 
3ft'  =  2m' I'. 

The  movable  needle,  being  drawn  in  one  direction  by  the 
earth's  magnetism  and  in  the  perpendicular  direction  by  the 
magnet,  takes  up  a  position  of  equilibrium  corresponding 
to  an  angle  a  between  its  axis  and  the  magnetic  meridian. 


$2  MAGNETISM. 

On  account  of  the  small  dimensions  of  the  needle,  we  may 
consider  that  the  forces  exerted  upon  its  poles  by  the 
poles  of  the  magnet  are  equal  and  opposite  and  form  a 
couple. 

The  couple  due  to  the  earth's  field  is  expressed  by 

w  —  3ft '  3C  sin  a. 

The  couple  due    to   the  magnet  is,  calling  F  the  force 
which  it  exerts  on  the  poles  of  the  needle, 

w'  =  2FI'  cos  a. 
Now  by  Coulomb's  law 

mm'  mm'  A.LI 

* 


"(£•_/•)' 

The  condition  of  equilibrium 

w  —  \ 
gives 


T    c*c")C  /y 

SOI  'X  sin  a  =  2SfnSfn  ' 


'_/«)»' 

whence 

3ft/          /Va_Z8tana; 

But 


if  /  is  sufficiently  small,  we  may  neglect  the  powers  of  ~ 
higher  than  the  second,  and  write 

!(< + 4} =n-- •  •  -  •  w 


PROPERTIES   OF  MAGNETS.  53 

In  general,  the  actual  distance  /  between  the  poles  is  un- 
known. This  term  may  be  eliminated  by  making  another 
experiment  on  the  deviation  of  the  needle  at  a  different  dis- 
tance. * 

We  get  an  angle  of  .fJeviatiotf  <*',  such  that 

3fft/  /2\       Lfi  tan  a' 


whence  by  subtracting  (3)  from  (2),  after  having  multiplied 
(2)  by  U  and  (3)  by  L'\   ' 

3TI       D  tan  a  —  L"  tan  a'  .  , 


OC  "  2<Za  -  L") 

Equations  (i)  and  (4)  permit  us  to  determine  separately 
the  magnetic  moment  of  the  magnet  and  the  horizontal 
intensity  of  the  earth's  field.  This  apparatus,  including  the 
magnetized  needle,  is  called  a  magnetometer,  because  it  en- 
ables us  to  compare  the  moment  of  two  magnets  or  the 
effects  of  a  single  magnet  placed  in  different  positions. 

49.  Remarks.  —  I.  The  needle  might  have  been  hung  in  a 
direction  normal  to  the  centre  of  the  magnet.     Adopting 
the    preceding    mode  of  calculation,  we  should   then   have 

obtained 

3fft       L*  tan  a  —  L'*  tan  a1 

5C  =  U  -  L" 

II.  The  results  obtained  in  similar  experiments  agree 
rigorously  with  the  calculated  results  based  on  Coulomb's 
law  and  furnish  the  proof  of  the  exactness  of  .this  law. 

50.  Measurement  Of  Angles.—  The  preceding  determi- 
nations necessitate  the  measurement  of  the  angles  of  equi- 
librium of  the  needle.     As  this  measurement  is  of  frequent 
occurrence  in  electrotechnics,  we  will  describe  some  of  the 
methods  of  readings  in  use. 


54 


MAGNETISM. 


The  unit  angle  employed  in  absolute  measurements  is  the 
radian,  or  arc  equal  to  the  radius  and  corresponding  to 


The  simplest  way  is  to  affix  to  the  needle  a  slender  pointer 
which  moves  over  a  horizontal  scale.  The  error  of  parallax 
is  avoided  by  placing  a  mirror  in  the  plane  of  the  scale  and 
putting  the  eye  of  the  observer  in  such  a  position  that  the 
pointer  coincides  with  its  reflection  in  the  mirror.  The 
reading  is  thus  made  in  a  plane  perpendicular  to  the  scale. 
This  mode  of  reading  does  not  allow  of  great  precision  ; 
when  the  angles  are  small,  the  relative  error  may  be  consid- 
erable. 

Greater  exactitude  is  obtained  by  reflection  methods 
recommended  by  Poggendorff  and  Lord  Kelvin. 

In  the  first,  called  subjective  method,  a  small  plane  mirror, 
M,  is  attached  to  the  axis  of  suspension  of  the  needle,  Fig. 
17,  and  at  a  certain  distance,  varying  from  3  to  10  feet,  is 


FIG.  17. 

placed  a  reading-telescope  with  micrometer  threads,  sup- 
ported on  a  tripod  with  adjustment  screws.  Above  and  be- 
low the  telescope  is  a  horizontal  scale,  graduated  decimally. 
Before  each  measurement  the  apparatus  is  adjusted  to  thas 


PROPERTIES   OF  MAGNETS.  55 

the  vertical  plane  through  the  optical  axis  of  the  telescope 
is  normal  to  both  the  scale  and  the  mirror.  To  do  this, 
sight  at  the  centre  of  the  mirror  and  move  the  telescope  and 
scale  until  the  division  of  the  scale,  which  lies  in  the  vertical 
plane  passing  through  the  optical  axis,  is  read  in  the  mirror. 
The  result  of  this  arrangement  is,  that  for  an  angle  of 
deviation  a  of  the  needle,  a  length  /  is  read  off  by  the  tele- 

scope, such  that 

/ 

tan  2a  =  Y, 
JLr 

L  being  the  distance  of  the  mirror  from  the  scale. 
Hence  we  deduce 

1  l  /        I  /  /        I    ?       I   /'          v 
a  =  —  tan"  —  =  —  I—  --  •—  _|_  —  T»  •••  )• 

2  L      2\L      3  U  '   5  D         ) 

If  the  angles  are  less  than  3°,  we  need  only  retain 

i   /       i  r 


This  calculation  may  be  avoided  by  using  a  scale  curved 
into  the  arc  of  a  circle  of  radius  L.  We  then  have  rigor- 

ously 

i    / 

a  =  2L- 

In  Sir  Wm.  Thomson's  objective  method  the  plane  mirror 
is  replaced  by  a  curved  one,  and  the  telescope  by  a  lamp 
which  emits  a  ray  of  light  through  a  diaphragm  furnished 
with  a  vertical  slit.  The  scale  is  placed  at  such  a  distance 
from  the  mirror  that  the  image  of  the  slit  is  cast  on  the 
scale-divisions.  These  may  be  marked  on  a  scale  made  of 
ground  glass,  tracing  cloth,  or  celluloid  ;  the  observer  stands 
behind  the  scale  and  sees  the  image  by  transparence.  When 
an  incandescent  lamp  is  used,  the  filament  of  the  lamp  gives 
an  extremely  precise  linear  image. 


56  MAGNETISM. 

The  movable  mirror  may  be  plane,  as  in  the  preceding 
case,  but  a  convergent  lens  must  then  be  placed  in  the  path 
of  the  ray  of  incident  light. 


INDUCED   MAGNETIZATION. 

51.  Magnetic  and   Diamagnetic   Bodies.  —  We     have 
seen,  §  39,  that  molecules  of  iron  placed  in  a  field  tend  to 
place  themselves  along  the  magnetic  lines  of  force.    The  dif- 
ferent varieties  of  iron,  cast  iron  and  steel  (with  the  excep- 
tion  of  manganese-steel),   together  with  cobalt  and  nickel, 
show  an  energetic  magnetization  in  a  magnetic  field.     Some 
other  bodies,  such  as  magnetic  oxide,  perchloride  and  sul- 
phate of  iron,  exhibit  the  same  properties,  but  to  a  much 
smaller  degree. 

Bismuth  is  also  magnetized  in  a  very  intense  field,  but  a 
bar  of  this  metal  tends  to  place  itself  perpendicularly  to  the 
lines  of  force.  In  sufficiently  powerful  fields  all  bodies  ex- 
hibit magnetic  properties,  but  to  an  incomparably  smaller 
extent  than  iron. 

Those  bodies  whose  magnetic  orientation  is  the  same  as 
that  of  iron  are  called  ferromagnetic  or  magnetic  ;  those 
which  act  like  bismuth  are  called  diamagnetic. 

52.  Coefficient  of  Magnetization  or  Magnetic  Suscep- 
tibility. —  The    problem    of    magnetizatign    by  induction 
amounts,  in  fact,  to  determining  for  the  various  parts  of  the 
magnets  the  ratio  of  the  intensity  of  magnetization  to  the 
field-intensity  or  magnetizing  force. 

This  ratio 

3 


is   called   the   coefficient   of  magnetization  or   the  magnetic 
susceptibility. 


INDUCED    MAGNETIZATION.  57 

It  is  easy  to  see  from  the  dimensions  of  3  and  3C,  §§  36, 
40,  that  this  coefficient  simply  represents  a  numerical  factor. 

If  an  isotropic  body,  whose  magnetic  susceptibility  is  the 
same  in  all  directions,  is  subniitted  to  a  magnetizing  force 
that  is  constant  in  jevery  psint  of  the  body,  it  tends  to 
acquire  a  constant  intensity  of  magnetization.  But  the 
magnetic  poles  induced  in  the  body  modify  the  field,  so  that 
if  the  field  were  uniform  before  the  introduction  of  the  body, 
it  becomes  non-uniform  in  consequence  of  that  introduction. 
It  is  very  difficult  in  the  majority  of  cases  to  determine  the 
resultant  field  and,  consequently,  the  actual  intensity  of  the 
magnetizing  force  in  every  point.  Thus  the  problem  of  the 
distribution  of  magnetism  in  a  short  cylinder,  whose  axis  is 
parallel  to  the  direction  of  the  field,  has  never  been  solved. 

Certain  cases,  however,  are  easily  calculated.  The  prac- 
tical way  of  obtaining  a  uniform  field  of  a  given  intensity 
consists,  as  will  be  seen  further  on,  in  sending  an  electric 
current  through  a  very  long  cylindrical  bobbin,  in  the  in- 
terior of  which  is  placed  the  body  to  be  magnetized. 

53.  Cases  of  a  Sphere  and  a  Disk. — Let  an  isotropic 
sphere  be  placed  in  a  uniform  field  of  intensity  CfC.  The 
various  elements  of  the  sphere  tend  to  assume  a  uniform 
magnetic  orientation,  in  consequence  of  which  there  are 
developed,  on  the  two  hemispheres  limited  by  the  great 
circle  normal  to  the  direction  of  the  field,  such  magnetic 
shells  that  the  resultant  internal  action  on  unit  pole  is  con- 
stant and  equal  to 

|*3 -     -     •;  ;•     (§42) 

The  intensity  of  the  field  in  the  interior  of  the  sphere  is 
therefore  constant  in  magnitude  and  direction  and  equal  to 

OC  —  -7r3; 


58  MAGNETISM. 

consequently 


whence 


> 

JC  -  -7T3 


AT3C 

0  :=  -- 


The  susceptibility  of  iron  is  always  much  higher  than 
unity.  It  follows,  then,  that  the  value  of  3  is  never  very 

TC  TP 

distant  from  -  =  -  .     Consequently  the  spherical  form 

—  7t 

3 

is  not  suitable  for  obtaining  high  intensities  of  magnetiza- 
tion. 

In  the  interior  of  an  infinitely  thin  disk,  transversely 
magnetized  in  such  a  way  that  the  magnetic  density  of  its 
faces,  equal  to  the  intensity  of  magnetization,  be  3,  the  com- 
ponent of  the  force  due  to  these  faces  is  (§  31) 


-f  2;r3  —  (—  2?r3)  = 
Consequently,  the  intensity  of  magnetization  becomes 

3  =  /c(3C  —  47T3), 
whence 


3   -f-  ^TTK 

a  value  which  tends  towards  — . 

The  transverse  magnetization  of  a  disk  of  iron  is  therefore 
always  very  feeble. 

The  same  can  be  shown  with  regard  to  the  magnetization 
of  an  iron  cylinder  in  a  direction  normal  to  its  axis. 


INDUCED   MAGNETIZATION.  59 

54.  Case  of  a  Ring. — A  ring  subjected  to  magnetizing 
forces,  constant  in   magnitude  and  directed  in  every  point 
of  the  ring  along  the  tangent  to  the  parallel  circle  passing 
through   that   point,  will  assume   a  constant  magnetization 
without  free   poles,  since  the^rows  of  magnetic  molecules 
will  form  closed  circular  chains. 

The  original  field  will  not,  therefore,  have  its  distribution 
modified  by  the  presence  of  the  ring,  and  the  intensity  of 
magnetization  will  be  simply  expressed  by 

3  =  /cOC, 

3C  representing  the  intensity  of  the  field. 

A  conductor  coiled  round  an  iron  ring,  and  traversed  by 
an  electric  current,  approximately  realizes  the  above  condi- 
tion, as  will  be  shown  later.  After  the  stoppage  of  the  cur- 
rent, the  annular  core  retains  the  greater  part  of  its  mag- 
netism in  the  permanent  state,  for,  in  the  absence  of  free 
poles,  there  is  no  demagnetizing  force. 

55.  Case  of  a  Cylinder  of  Indefinite  Extent. — A  third 
solution    is   furnished   by  a    cylinder  of   indefinite   extent, 
placed  parallel  to  the  lines  of  force  of  a  uniform  field.     The 
intensity  of  the  field  in  the  interior  of  the  cylinder  is  the  re- 
sultant of  the  original  field  and  of  the  action  of  the  poles 
induced  at  the  extremities  of  the  cylinder. 

The  longitudinal  magnetization  that  can  be  given  to  an 
iron  cylinder,  whose  axis  is  placed  parallel  to  the  direction 
of  the  field,  increases  as  the  length  of  the  cylinder  is  in- 
creased, since  the  effect  of  its  poles  then  produces  less  and 
less  diminution  of  the  intensity  of  the  field  inside  the  cylin- 
der. Short  steel  cylinders  cannot,  therefore,  make  good 
permanent  magnets,  for  they  become  only  slightly  magnet- 
tized,  and  the  reaction  of  the  poles  tends  to  rapidly  change 
the  molecular  orientation  after  taking  away  the  magnetizing 


60  MAGNETISM. 

force.     On  the  other  hand,  long  cylinders  become  strongly 
magnetized  and  retain  their  magnetism. 

When  an  iron  cylinder  is  of  indefinite  length,  the  demag- 
netizing action  of  its  poles  becomes  negligible  for  points 
situated  in  the  accessible  region  of  the  cylinder  where  the 
intensity  of  magnetization  is  uniform  and  expressed  by 

3  =  xrOC. 

It  has  been  shown  experimentally  that  this  formula  is 
still  applicable  when  the  length  of  the  cylinder  is  equal  to 
400  or  500  times  its  diameter. 

56.  Portative  Power  of  a  Magnet.  —  Let  us  consider  a 
cylinder  of  indefinite  length,  magnetized  parallel  to  its  axis. 
If  we  imagine  a  narrow  crevasse  cut  out  normal  to  the  axis, 
the  opposite  walls  will  be  covered  with  magnetic  masses 
whose  density  is  equal  to  the  intensity  of  magnetization, 

(7  =  0. 

The  force  with  which  unit  mass,  situated  near  the  face 
whose  density  is  —  cr,  is  attracted  by  this  latter,  is  expressed 
by  27T<r,  §  31. 

Consequently  the  mass  -f-  cr,  which  covers  unit  surface  on 
the  opposite  wall,  is  attracted  with  a  force 

27T(7a  ==  27T3'. 

This  is  the  portative  force  of  the  magnet  per  unit  surface. 

If,  moreover,  the  cylinder  is  under  the  action  of  a  field  of 
intensity  3C,  —  that  is,  a  field  capable  of  exercising  a  force  3C 
on  unit  pole,  and  directed  parallel  to  3,  —  the  portative  force 
must  be  increased  by 

ae  o-  = 


The  total  portative  force  will  then  be 
OC3       27T32 


INDUCED   MAGNETIZATION.  6  1 

per  unit  surface,  or 


for  a  surface  5. 

These  expressions  furnish  a  simple  means  of  determining 
the  intensity  of  magnetization.  In  the  first  case,  it  is  only 
necessary  to  apply  weights  to  one  of  the  portions  of  the 
cylinder  until  it  is  no  longer  able  to  sustain  them  ;  let  w 
dynes  be  the  weight  at  which  it  ceases  to  sustain, 


=  w 
whence 

w 
3  = 


' 

Some  authors  give  the  following  expression  for  the  porta- 
ve  power : 


tive  power : 

OC2 
— -. 

O7T 

The  last  term  relates  to  the  force  needed  to  separate  the 
magnetic  field  itself  into  two  parts.  It  is  the  force  neces- 
sary to  separate  into  two  portions  an  indefinitely  long  mag- 
netizing bobbin,  for  we  shall  see  that  the  mass  of  the  poles 
of  such  a  bobbin  is  represented,  per  unit  surface,  by  #,/, 
n^  being  the  number  of  turns  per  centimetre,  and  /the  current. 
The  portative  power  of  the  bobbin  is  consequently  2nn?I*  = 

rtf)  /TT>2 

5 — ,  for  OC  =  ^nnj.     The  term  ~  is,  moreover,  very  small  in 

O/T  O7T 

the  case  of  an  iron  core. 

57.  Variations  of  Intensity  of  Magnetization  with 
the  Magnetizing  Force.  Hysteresis.* — When  an  indefi- 
nitely long  iron  bar,  in  the  neutral  state  and  softened  by 
annealing,  is  subjected  to  the  influence  of  a  field  of  increas- 

*See  also  chapter  on  Hysteresis,  by  Charles  Proteus  Steinmetz,  p.  87. 


62 


MAGNETISM. 


ing  intensity,  the  magnetization  3  varies  as  shown  by  the 
curve  OA,  Fig.  18,  whose  abscissae  represent  the  values  of  3C. 
We  see  that  for  very  small  forces  the  magnetization  increases 
slowly.  Beyond  3C  =  i  C.  G.  S.  unit,  the  curve  shows  a 
point  of  flexion  beyond  which  the  ordinates  grow  rapidly 
larger  up  to  a  point  corresponding  to  values  of  3C  comprised 
between  5  and  10  C.  G.  S.  units,  where  the  curve  makes  a 
sharp  turn.  The  increase  of  the  ordinates  then  becomes 
smaller  and  smaller  and  the  bar  reaches  the  state  commonly 
known  by  the  name  of  saturation,  which  corresponds  rigor- 
ously to  the  ordinate  of  a  horizontal  asymptote  to  which  the 
curve  approaches  indefinitely.  The  magnetizing  forces 


FIG.  18. 

shown  here  are  obtained  in  practice  by  placing  the  bar  in  a 
very  long  solenoid  traversed  by  an  increasing  current.  The 
intensity  of  the  field  inside  the  solenoid  is  proportional  to 
the  current. 

According  to  Ewing  and  Low,  the  intensities  of  magnet- 
ization corresponding  to  saturation  are  approximately,  in 
C.  G.  S.  units,  for  wrought  iron  1700,  for  cast  iron  1240,  and 
for  nickel  513. 


INDUCED    MAGNETIZATION.  63 

Steel,  which  can  attain  the  same  magnetization  as  iron 
under  very  powerful  magnetizing  forces,  scarcely  retains 
more  than  half  in  the  form  of  permanent  magnetism. 

The  magnetization-curve  >sfiows  that  the  susceptibility 
3 

K  =  —  is  at  first  very  feeble ;  then  it  increases  rapidly  with 
CJC 

3C  and  attains  for  iron  a  value  varying  from  200  to  300.  It 
then  decreases  progressively  to  a  very  small  value. 

For  slightly  magnetic  bodies  the  susceptibility  is  always 
less  than  o.ooooi  in  absolute  value  ;  in  these  conditions  it 
may  practically  be  considered  as  zero  compared  to  the 
susceptibility  of  iron,  for  moderate  magnetizing  forces. 

Let  us  suppose  that  the  bar,  after  having  reached  the 
point  A  corresponding  to  saturation,  be  submitted  to  mag- 
netizing forces  decreasing  from  OB  to  zero.  The  mag- 
netization does  not  pass  again  through  the  intermediate 
states  first  observed,  but  varies  according  to  the  curve  AC, 
OC  corresponding  to  the  residual  magnetism. 

If  the  magnetizing  force  changes  its  direction  and  takes 
a  negative  value  OB' ,  equal  to  OB,  the  intensity  takes  the 
successive  values  shown  by  the  curve  CA '.  Finally,  the 
magnetizing  force  repassing  through  the  consecutive  values 
between  B'  and  B,  the  magnetism  of  the  bar  will  return  to 
the  value  AB  by  a  curve  A'C'A. 

The  cycle  AC  A'C'A  can  be  reproduced  indefinitely  by 
causing  the  field-intensity  to  vary  periodically  between  the 
values  OB  and  OB' ,  which  is  obtained  by  giving  to  the 
current  traversing  the  magnetizing  solenoid  values  oscillat- 
ing between  two  equal  limits  of  contrary  sign.  The  curves 
joining  the  points  At ,  AJ,  Fig.  18,  show  the  variations  of 
the  magnetic  state  of  the  same  iron  bar  hardened  by  the 
application  of  a  tractive  force  greater  than  the  limit  of 
elasticity  of  the  metal.  It  will  be  observed  that  the  maxi- 
mum magnetization  is  less  than  when  the  metal  is  annealed. 


64  MAGNETISM. 

Moreover,  the  susceptibility  of  the  hardened  metal  is  con- 
siderably diminished. 

It  will  be  seen  by  the  a.bove  that  the  magnetization  of  a 
bar  is  capable  of  assuming  very  different  values  for  the  same 
magnetizing  force ;  it  depends  not  only  on  the  actual  mag- 
netizing force,  but  also  on  the  preceding  magnetic  condi- 
tions. The  intensity  of  magnetization  of  an  iron  core  is  a 
complex  function  of  the  magnetizing  force  and  the  preced- 
ing condition  of  the  iron. 

During  the  period  of  decrease  in  the  cyclic  curve,  the 
values  of  the  intensity  of  magnetization  are  always  larger 
than  those  given  by  the  curve  OA,  while  during  the  period 
of  increase  they  are  smaller.  This  phenomenon,  due  to  the 
coercive  force,  has  been  called  by  Ewing  hysteresis  (from  the 
Greek,  lagging  behind}* 

The  ordinate  at  the  origin,  OC,  represents  the  residual 
magnetism  of  the  bar.  When  soft  iron  is  kept  from  all 
vibrations,  this  ordinate  equals  nearly  three  fourths  of  the 
maximum  ordinate  of  the  curve. 

Dr.  Hopkinson  has  especially  designated  by  the  name 
coercive  force  the  magnetizing  force,  OD,  which  must  be  ap- 
plied to  the  bar  (in  reverse  direction)  to  destroy  its  residual 
magnetism.  The  bar  is  then  not,  however,  in  the  neutral 
state,  for  it  has  a  very  different  susceptibility  from  that  ob- 
served on  beginning  the  magnetization  ;  for  it  is  readier  to 
take  a  negative  magnetization  than  when  it  was  in  the  neu- 
tral state.  In  the  hardened  metal  the  coercive  force,  ODV1 
is  appreciably  increased. 

The  form  of  the  curve  ACA'C  shows  that  to  bring  a  bar 
back  to  the  neutral  state  it  must  be  subjected  to  periodic 
forces  of  decreasing  intensity.  The  cycles  then  described  will 
approach  nearer  and  nearer  to  the  origin.  It  is  for  this  reason 

*  See  Ewing,  Magnetic  Induction  in  Iron,  etc.,  p.  93  et  seq. 


INDUCED   MAGNETIZATION.  65 

that  to  demagnetize  a  watch  it  must  be  placed  in  the  field 
of  a  magnet  and  then  steadily  drawn  away,  making  it  at 
the  same  time  revolve  in  order  to  change  the  direction  of 
the  magnetizing  force,  which  grows  feebler  as  the  distance 
from  the  magnet  increases. 

Figure  19  is  intended  to  show  the  difference  existing  be- 
tween the  neutral  state  and  that  of  zero  magnetization.  A 
bar  in  the  neutral  state  has  been  subjected  to  increasing 
magnetizing  forces,  bringing  it  up  to  the  magnetization  A. 
The  field  has  then  been  gradually  reduced  to  zero  and  next 
changed  in  direction.  At  a  certain  point,  a  momentary 


FIG.  19. 

return  of  the  magnetizing  force  to  zero  has  produced  the 
loop  BC\  then  the  values  of  the  field  have  recommenced  to 
decrease.  A  new  retrogression  has  occurred  at  Dy  chosen 
so  that  the  curve  of  magnetization  passes  through  the 


66  MAGNETISM. 

origin.  At  the  moment  of  passing  through  the  origin,  the 
bar  is  not  in  the  neutral  state,  for  if  the  magnetizing  force 
be  increased,  the  curve  continues  along  OE  and  not  along 
OA.  On  completing  the  cycle  of  the  field-intensity,  we 
return  to  a  point  which  does  not  coincide  with  A  unless  this 
latter  corresponds  to  the  point  of  saturation  of  the  bar. 

58.  Frolich's  Formula. — Various  writers  have  tried  to 
represent,  by  empirical  formulae,  the  variation  of  the 
intensity  of  magnetization  as  a  function  of  the  magnetizing 
force. 

If  we  suppose  that  the  susceptibility  is  proportional  to 
the  difference  between  the  maximum  intensity  of  magnetiza- 
tion and  the  actual  intensity,  we  have 

x  =  ~  =A(3m-Q, 

whence 

A  3m3C  a  OC 


3  = 


a  and  b  being  constants  for  a  given  bar. 

This  curve  represents  an  hyperbola  passing  through  the 
origin  and  one  of  whose  asymptotes  is  parallel  to  the  axis 
of*. 

By  choosing  the  parameters  a  and  b  suitably,  we  can,  for 
approximate  calculations,  substitute  this  curve  for  the  true 
one  obtained  by  experiment.  Frolich  and  S.  P.  Thomp- 
son have  applied  this  formula  to  the  theory  of  dynamos. 

59.  Muller,  von  Waltenhofen,  and  Kapp  have  adopted  a 
formula  of  the  form 

3  —  a  tan-1  £OC, 

which  can  likewise  furnish  approximate  values  of  intensities 
of  magnetization  of  soft  iron  by  a  suitable  choice  of  param- 


INDUCED    MAGNETIZATION.  67 

eters.  This  formula  is  not  so  convenient  in  calculating  as 
the  preceding  one.  It  will  be  noticed  that  the  above  equa- 
tions represent  curves  passing  through  the  origin  and  that 
they  consequently  leave  out  ,th"e  phenomenon  of  hysteresis. 
They  give,  at  the  most,  a  curse  intermediate  between  the 
two  curves  obtained  in  a  magnetic  cycle.* 

60.  Another  Way  of  Looking  at  Induced  Magnetiza- 
tion,  Magnetic    Induction,   and   Permeability.  —  Let   us 

consider  a  uniform  field,  in  which  the  intensity  3C  represents 
the  flux  of  force  across  unit  equipotential  surface,  which  is 
measured  by  the  force  exercised  on  unit  pole.  If  we  place 
an  indefinitely  long  cylinder  parallel  to  the  direction  of  the 
field,  the  space  occupied  by  the  cylinder  becomes  the  seat 
of  a  different  flux,  that  is,  the  force  exercised  on  a  unit  pole, 
hypothetically  placed  inside  the  cylinder,  is  modified  in  a  way 
that  will  appear  later  on.  This  flux,  (B,  per  unit  section  is 
sometimes  called  the  magnetic  induction  across  the  cylinder. 

*  Drs.  Houston  and  Kennelly  have  recently  pointed  out  that  from  the 
researches  of  Ewing,  Klaassen,  Fessenden,  and  others 

(i)  An  approximate  linear  relation  exists  between  remanance  and  maxi- 
mum cyclic  intensity  in  iron  and  steel,  or,  symbolically, 


where  (B0  is  the  remanance  or  residual  cyclic  magnetic  flux  when  the  mag- 
netizing force  is  zero,  and  (Bmax.  is  the  maximum  cylic  intensity.  This  can 
only  be  regarded  as  an  empirical  formula,  holding  between  (Bmax.  =  500  and 
(Bmax.  =  9000,  in  some  cases  up  to  (Bmax.  =  16,000. 

(2)  An  approximate  linear  relation  exists  between  coercive  force  and  maxi- 
mum cylic  intensity  in  iron  and  steel;  or,  symbolically, 

JCo  —  a\  -\-  ^i(Bmax. 

where  3C0  is  the  cyclic  coercive  force  or  the  value  of  JC  at  which  (B  =  o,  and 
(Bmax.  is  the  maximum  cyclic  intensity.  This  can  only  be  regarded  as  an 
empirical  formula,  holding  above  (Bmax.  =  4000. 

(3)  As  a  consequence  of  the  preceding   relations  an  approximate  linear 
relation  exists  between  remanance  and  coercive  force  in  the  cyclic  magnet- 
ization of  iron  and  steel  between  the  limits  above  denned. 


68  MAGNETISM. 

The  ratio 


between  the  magnetic  induction  and  the  magnetizing  force 
is  the  coefficient  of  permeability  (or  simply  permeability)  of 
the  cylinder.  It  follows  from  this  definition  that  the  perme- 
ability of  the  medium  into  which  the  substance  is  introduced, 
generally  the  air,  is  taken  as  unity. 

The  permeability  depends  on  the  nature  of  the  substance 
and,  in  highly  magnetic  bodies,  on  the  intensity  of  the  field. 
The  values  of  (B  and  /t  may  be  determined  directly  by 
electric  measurements.  These  quantities  are  rendered  im- 
portant by  the  fact  that  they  are  connected,  by  a  simple 
relation  with  the  intensity  of  magnetization  and  the  suscep- 
tibility. 

In  order  to  estimate  the  flux  of  force  inside  the  cylinder, 
let  us  suppose,  as  in  §  56,  that  an  infinitely  narrow  crevasse 
be  cut  in  the  cylinder  perpendicularly  to  its  axis.  We  can 
consider  that  this  operation  does  not  modify  the  total  flux 
across  the  cylinder.  The  walls  of  the  crevasse  normal  to 
the  direction  of  the  cylinder  are  charged  by  induction  with 
free  magnetism  having  densities  -{-  cr  and  —  o%  such  that 

a  =  3, 

3  being  the  intensity  of  magnetization  of  the  cylinder. 

The  effect  of  these  surface-charges  on  unit  pole,  supposed 
to  be  introduced  into  the  middle  of  the  crevasse,  is  to  pro- 
duce two  components  in  the  same  direction,  equal  to 

27TCT  =  27f3, 

having  a  direction  parallel  to  the  axis  of  the  cylinder,  §  31. 
Besides  this  we  must  add  to  these  components  the  force  3C 
due  to  the  field.  Since  the  two  components  are  parallel  by 
hypothesis,  the  resultant  will  be 


INDUCED   MAGNETIZATION.  69 

<B  =  oe  +  47T3  =  Oe(i+47r/c) (2) 

Comparing  equations  (i)  and  (2),  we  see  that 

/i  =i»4  4** (3) 

It  follows  from  the  preceding  equations  that  we  can 
express  the  magnetization  of  a  body  in  a  field  by  the 
intensity  of  magnetization  or  by  the  magnetic  induction 
indifferently.  It  seems  at  first  sight  that  one  of  these 
expressions  is  superfluous,  and  that  the  use  of  both  can 
only  produce  a  confusion  of  ideas.  But,  as  will  be  seen 
more  clearly  further  on,  there  are  cases  where  it  is  more 
convenient  to  use  the  first  expression,  and  other  cases  where 
the  second  expresses  the  phenomena  more  clearly.  Thus, 
when  we  consider  a  magnetized  bar,  the  moment  of  which 
is  determined  by  the  magnetometer,  §  48,  the  intensity  of 
magnetization  is  expressed  by  the  ratio  of  its  moment  to  its 
volume.  But  in  the  case  of  a  ring,  §  54,  in  which  the  lines 
of  force  are  closed,  the  external  magnetic  effect  would  be 
zero,  as  would  the  moment  also,  for  the  ring  has  no  free 
pole  and  its  magnetism  cannot  be  called  into  action  except 
by  making  a  section  in  a  plane  passing  through  the  axis  of 
revolution.  The  walls  thus  exposed  present  poles  of  con- 
trary name  and  whose  density  represents  the  intensity  of 
magnetization  of  the  body.  Between  these  poles  a  uniform 
field  is  developed  in  which  there  is  a  flux  equal  to  4713  per 
unit  of  section  and  which  is  to  be  added  to  the  flux  3C  in 
the  same  direction  due  to  external  causes.  It  will  be  seen 
in  the  part  on  Electromagnetism  that  the  total  flux,  called 
magnetic  induction  across  the  ring,  is  capable  of  being  deter- 
mined directly.  The  intensity  of  magnetization  is  deduced 
from  this  quantity  by  subtracting  the  value  of  3C  and  divid- 
ing the  remainder  by  4^-. 

Some  authors  estimate  the  magnetism  of  magnetized  bars 


?0  MAGNETISM. 

in  units  of  magnetic  induction.  In  this  case  it  is  only 
necessary  to  multiply  by  4?r  the  mean  value  of  the  intensity 
of  magnetization  found  by  means  of  the  magnetometer. 

While,  in  the  case  of  a  ring,  the  flux  of  magnetic  force 
remains  in  the  iron,  in  the  case  of  a  straight  magnet  the 
flux  leaves  the  iron  and  is  closed  through  the  surrounding 
air.  In  the  first  example  the  medium  is  homogeneous,  in 
the  second  heterogeneous,  being  composed  partly  of  iron 
and  partly  of  air;  but  in  both  cases  the  flux  should  be  con- 
sidered as  continuous  and  closed  on  itself.  This  way  of 
looking  at  the  matter  has  helped  to  simplify  the  conception 
of  magnetic  phenomena.  •  It  will  appear  in  the  course  of 
this  work  how  much  has  been  gained  by  extending  to  cir- 
cuits traversed  by  magnetic  fluxes  the  conditions  shown  to 
exist  for  circuits  traversed  by  electric  currents. 

By  definition,  the  permeability  of  air  is  unity.  Experi- 
ments show  that  the  permeability  of  a  vacuum  is  sensibly  of 
the  same  value.  Magnetic  bodies  are  those  of  which  the 
permeability  exceeds  that  of  air ;  diamagnetic  bodies,  those 
of  which  the  permeability  is  less  than  that  of  air.  This  can 
also  be  expressed  in  the  statement  that  magnetic  bodies 
conduct  lines  of  force  more  readily,  and  diamagnetic  bodies 
less  readily,  than  air. 

From  the  relation 

^  =  47T/C  +   I 

we  have 


K  = 


This  shows  that  the  susceptibility  of  magnetic  bodies  for 
which/<>i,  is  superior  to  zero,  while  the  susceptibility  of 
diamagnetic  bodies  is  negative. 

The  susceptibility  and  permeability  of  iron,  cobalt  and 
nickel  at  ordinary  temperatures  are  so  superior  to  those  of 


INDUCED    MAGNETIZATION.  7 1 

other  bodies,  whether  magnetic  or  diamagnetic,  that  there  is 
no  practical  error  in  taking  the  permeability  of  all  other 
bodies  as  equal  to  unity  and  their  susceptibility  as  zero. 
The  most  diamagnetic  body  in  existence,  bismuth,  has  a 
permeability  of  0.9991^ 

61.  Work  spent  in  Magnetizing. — As  the  magnetiza- 
tion of  a  body  imparts  to  it  a  certain  quantity  of  potential 
energy,  it  necessitates  a  certain  expenditure  of  work.  We 
shall  show  later  on  that  this  work  is  expressed,  per  unit 
volume  of  the  magnetized  body,  by 


i      C  l     C 

-    I  OCd(B=r —   /  yuOC 

^J  ^J 


the  integral  being  extended  to  the  limits  between  which  the 
induction  of  the  magnet  has  been  changed.  If  the  values 
of  the  induction  compared  with  the  field-intensity  be  shown 
by  the  curve  OA,  Fig.  20,  and  if  the  magnetization  reach 
the  state  denoted  by  the  point  A,  the  work  expended  will 


FIG.  20. 


be  represented  by  the  area,  divided  by  471-,  of  the  surface 
comprised  between  the  curve,  the  axis  of  ordinates  and  a 
parallel  to  the  axis  of  abscissae  drawn  through  A, 


72  MAGNETISM, 

If  jj.  were  a  constant  factor,  as  is  the  case  for  slightly 
magnetic  substances,  the  integral  would  reduce  to 


for  a  variation  extending  between  o  and  3C. 

Let  us  suppose  that  an  indefinitely  long  bar,  after  having 
reached  the  magnetic  state  A,  traverses  a  cycle  ACA'C'A, 
Fig.  20,  the  intensity  of  the  field  passing  from  the  value  OB 
to  o,  and  then  returning  to  OB. 

The  integral 

/»&=  AB 

—  I  oed& 

4^<B  =  04' 

representing  the  area  of  the  surface  A'  C'  AC  divided  by  4?r, 
expresses  the  work  expended  in  order  to  cause  unit  volume 
of  the  bar  to  pass  through  the  given  cycle.  This  work  is 
transformed  into  heat  in  the  substance,  and  is  the  loss  due 
to  hysteresis. 

In  the  case  where  the  cycle  through  which  the  magnet- 
ized bar  goes  is  produced  by  forces  oscillating  between 
values  OB  and  OB'  ,  Fig.  21,  the  loss  by  hysteresis  is  repre- 
sented by  the  area  ACA'C'A. 

In  the  case  of  a  completed  cycle  the  expression  for  the 
energy  dissipated  is  susceptible  of  a  simpler  expression. 
Thus,  replacing  (&  by  ^n  3  +  5C,  we  get 


d3C. 


Now   the   second   integral   is  cancelled  for  a  completed 
cycle  and  the  energy  is  then  expressed  by 


/OC  d<B  =  /X.  d3  +  i 


INDUCED   MAGNETIZATION.  J  $ 

It  is  readily  seen  that  the  loss  increases  with  the  coercive 
force,  represented  by  the  abscissa  at  the  origin  of  the  curve 
ACAf,  and  with  the  maximum  intensity  of  magnetization. 

When  a  magnet  is  in  a  state  "of  repose  the  loss  is  greater 
than  in  the  dynamic  ^state,  especially  for  soft  iron,  for  we 
have  seen  that  the  coercive  force  is  diminished  by  vibrations 
of  the  substance. 


FIG.  21, 

When  the  cycle  of  the  magnetizing  force  is  completed  very 
rapidly,  the  magnetization  attained  by  the  metal  is  dimin- 
ished, as  well  as  the  loss  by  hysteresis.  Thus  M.  Tanaka- 
date  has  found  that  when  the  duration  of  the  cycle  is  be- 
tween -fa  and  fl^  of  a  second,  the  loss  is  only  eight  tenths 
of  that  observed  in  the  case  of  cycles  completed  more 
slowly. 


74  MAGNETISM. 

62.  Numerical  Results. — The  preceding  equations  show 
that  the  values  of  the  permeability  of  an  iron  bar  pass 
through  variations  analogous  to  those  of  its  susceptibility. 
At  first  very  small  for  small  values  of  the  magnetizing  force, 
the  permeability  grows  rapidly  towards  a  maximum,  then 
decreases  indefinitely  towards  a  value  but  slightly  differing 
from  that  of  air. 

The  curves,  Figs.  18  and  19,  showing  the  variations  of 
intensity  of  magnetization  of  a  bar  in  terms  of  the  mag- 
netizing force,  also  show,  very  sensibly,  the  magnetic  induc- 
tion referred  to  this  same  force  if  we  regard  unity  on  the 
scale  of  the  ordinates  as  representing  the  number  471-. 

Below  is  a  table  of  magnetic  values  as  found  for  two 
specimens  of  annealed  soft  iron,  and  one  of  gray  cast-iron. 


Annealed  Soft  Iron. 

Gray  Cast-iron. 

(B 

M- 

<£ 

V 

(B 

V 

1,000 

560 

11,000 

1,692 

4,000 

800 

2,000 

880 

12,000 

1,412 

5,000 

500 

3.000 

1,160 

13,000 

1,083 

6,000 

279 

4,000 

1,400 

14,000 

823 

7,000 

133 

5,000 

i,  600 

15,000 

526 

8,000 

100 

6,000 

1,800 

16,000 

320 

9,000 

71 

7,000 

1,960 

17,000 

161 

10,000 

53 

8,oco 

2,120 

i8,oco 

90 

11,000 

37 

9,000 

2,280 

19,000 

54 

10,000 

2,000 

20,000 

30 

Very  pure  soft  iron  shows  the  greatest  permeability  of 
any  metal.  It  is  followed,  in  descending  order,  by  soft 
steels  (Thomas  and  Bessemer),  malleable  iron,  and  gray  cast- 
iron,  the  magnetic  qualities  of  which  are  very  variable  ac- 
cording to  its  composition.  Tempered  steel,  in  which  the 
iron  is  in  an  especial  condition,  has  a  very  low  permeability, 
while  steel  containing  12  per  cent,  of  manganese  is  hardly 
more  magnetic  than  air. 


INDUCED    MAGNETIZATION. 


75 


The  following  curves  represent  the  permeability  as  a 
function  of  the  magnetic  induction  for  various  specimens  of 
iron  experimented  on  by  Rowland,  Hopkinson,  and  Bid- 
well. 

The  coercive  force,  measure&by  the  abscissa  at  the  origin 
of  the  curve  of  magnetism,  is  only  about  2  for  soft  iron  ;  it 


FIG.  22. 

reaches  40  for  chrome-steel  tempered  in  oil,  and  50  for  steel 
containing  3  to  4  per  cent,  of  tungsten.  This  result  shows 
that  the  latter  steel  is  especially  suitable  for  making  per- 
manent magnets. 

The  energy  dissipated  by  hysteresis  is  in  proportion  to 
the  maximum  induction  to  which  the  metal  is  subjected  and 
to  its  coercive  force.  While  this  energy  varies  between 


76 


MAGNETISM. 


10,000  and  15,000  ergs  for  specimens  of  iron  and  soft  steel 
subjected  to  magnetizing  forces  oscillating  between  values 
high  enough  to  produce  saturation,  it  attains  216,000  ergs 
in  tungsten-steel  (Hopkinson). 

The  permeability  of  annealed  soft  iron  decreases  very 
nearly  in  proportion  to  the  increase  of  (B,  between  values 
corresponding  to  <$>  =  7  kilogausses  and  (B  =  16  kilogausses. 
Between  these  limits  the  mean  values  of  the  permeability 
are  approximately  given  by  the  empirical  formula,  deduced 
from  Hopkinson's  curve, 

*  =  ~~  3^5  +  48S°* 

Figure  23  presents  a  curve  determined  by  Ewing  and  show- 
ing the  loss  in  a  soft-iron  bar  subjected  to  increasing  alter- 
nating magnetizing  forces.  The  ordinates  of  the  curve 
denote  ergs  per  cm3  and  the  abscissae  give  in  C.  G.  S.  units 
the  extreme  values,  positive  and  negative,  of  the  magnetic 
induction  through  the  metal. 


w 

10000 


8000 
6000 
4000 
2000 
0 


FIG.  23. 

According  to   Steinmetz,  the  loss  of  energy  in  ergs  is 
represented  by  the  expression* 

w  —  77  (B1-6, 


*  Steinmetz,  L' Industrie  Jlectrique,  March  6,  1892. 


INDUCED   MAGNETIZATION. 


77 


where  77,  the  coefficient  of  hysteresis,  may  have  the  following 
values : 


Material. 

Composition  and  State. 

Coefficient 
of 
Hysteresis, 
i? 

Anneal 

•045  P 
.032 
.032 
4-73 

3-35 
3-45 

ed 

ercent.  c 

i        < 

<        < 
«        « 

«        • 

>f  carbon,  annealed 
«               « 

"        tempered 
manganese,  forged 
tungsten,  tempered 
carbon 

.00202 
.OO262 
.00598 
.00954 
.05963 
.05778 
.01826 

Soft  Bessemer  steel  .  •  •  • 

Manganese-steel.  .  .  •      • 

Gray  cast-iron  

With  the  magnetizing  forces  that  can  be  obtained  in 
dynamo-electric  machines,  their  iron  cores  seldom  exceed 
an  induction  of  20,000  C.  G.  S.  units,  or  gausses,  but  by  estab- 
lishing particularly  powerful  fields  Messrs.  Ewing  and  Low 
have  succeeded  in  communicating  to  very  soft  iron  an  in- 
duction of  45  kilogausses.  Under  high  inductions,  the  in- 
tensity of  magnetization  has  a  constant  value  of  about  1700 
C.  G.  S.  units,  corresponding  to  saturation,  and  the  per- 
meability falls  to  a  constant  value  of  between  I  and  2. 

We  then  get  the  relation 

&:=3e-|-47r3  =  3e-|-  constant. 

In  very  intense  fields  cobalt  is  capable  of  attaining  the 
same  maximum  intensity  of  magnetization  as  cast-iron,  or 
about  three  fourths  of  the  magnetization  of  soft  iron.  Nickel 
never  exceeds  one  third  of  the  maximum  intensity  of 
magnetization  of  soft  iron. 

Lord  Rayleigh  has  found  that  in  very  weak  fields  the  per- 
meability can  be  expressed  by  a  formula 


=  a 


b  3C. 


For  a  specimen  of  soft  iron  he  has  found  #=8i  and  £=64. 
He  has  also  established  the  fact  that  hysteresis  is  absent 


7  MAGNETISM. 

when  a  bar  is  subjected  to  magnetizing  forces  varying  be- 
tween very  narrow  limits,  whether  the  metal  already  possesses 
any  magnetization,  or  if  it  has  been  taken  in  the  neutral 
state;  in  these  conditions  the  permeability  is  constant. 
When  these  small  variations  occur  near  a  magnetizing  force 
of  29  C.  G.  S.  units  or  gilberts,  he  has  found  that  the  per- 
meability of  soft  iron  is  only  80  per  cent,  of  the  permeability 
near  the  neutral  state. 

63.  Effect  of  Temperature  on  Magnetism.  Recales- 
cence. — We  have  already  made  allusion  to  the  influence  of 
the  temperature  on  the  magnetism  of  iron  and  its  deriva- 
tives, steel  and  cast-iron,  whose  magnetism  disappears  com- 
pletely at  a  bright  red  heat. 

Dr.  Hopkinson,  to  whom  we  are  indebted  for  precise 
experiments  on  this  thermic  effect,*  has  observed  that  in  a 
feeble  and  constant  field  of  0.3  gauss  the  permeability  of 
a  soft-iron  bar,  heated  gradually,  increases  progressively 
from  500  to  11,000;  but  at  the  temperature  of  775°  C.,  the 
permeability  falls  suddenly  to  a  value  very  close  to  I. 

When  the  intensity  of  the  field  increases,  the  increase  of 
permeability  is  much  less  sensible  and  the  fall  is  less  sudden. 
Finally,  in  an  intense  field,  the  permeability  decreases  con- 
tinuously with  the  rise  of  temperature.  In  every  case  the 
iron  becomes  completely  demagnetized  at  a  temperature 
in  the  neighborhood  of  785°  C.;  Dr.  Hopkinson  calls  this 
thermic  point  the  critical  temperature  of  the  metal. 

For  exceptionally  soft  iron  the  critical  temperature  may 
rise  as  high  as  880°  C.,  while  in  steel  it  falls  to  690°.  For 
nickel  the  critical  temperature  is  about  310°  C. 

The  critical  temperature  seems  to  correspond  to  a  mo- 
lecular change  in  the  substances,  shown  likewise  by  other 

*  See  Hopkinson,  Magnetism,  Journal  of  the  Institution  of  Electrical 
Engineers,  vol.  xix. 


INDUCED    MAGNETIZATION.  79 

phenomena.  Kohlrausch  has  observed  that  at  this  temper- 
ature the  electrical  -resistance  of  iron  shows  a  sudden 
variation. 

According  to  Tait  the  tfyeftno-electric  power  of  iron  is 
also  modified  in  a  profound  degree  towards  this  point ;  and 
lastly,  Barrett  has  discovered  a  very  characteristic  effect,  to 
which  he  has  given  the  name  of  recalescence.  If  we  allow  a 
piece  of  iron  or  steel  to  cool  down  after  having  heated  it  to 
a  bright  red,  there  comes  a  certain  stage  where  the  process 
of  cooling  stops  and  where  the  piece  becomes  slightly 
heated  again,  after  which  the  decrease  of  temperature  goes 
on  again  regularly.  This  recalescence  is  shown  in  hard  steel 
by  a  very  visible  luminous  effect,  the  color  of  the  metal 
passing  from  a  dull  red  to  very  bright  red  at  the  moment 
when  the  critical  temperature  is  reached.  This  experiment 
succeeds  very  well  when  a  knitting-needle  is  used,  first  heat- 
ing it  to  a  bright  red  by  passing  an  electric  current  through  it. 

It  is  very  surprising  that  magnetic  qualities  should  be 
clearly  exhibited  by  only  three  metals — iron,  nickel,  and 
cobalt.  The  other  elementary  bodies  are  so  little  capable 
of  magnetization  that  they  are  ordinarily  considered  as  non- 
magnetic. It  may  be  that  it  is  only  a  mere  question  of 
temperature,  the  three  metals  mentioned  being  the  only 
ones  which  manifest  decided  magnetic  properties  at  the 
ordinary  temperatures.  This  was  Faraday's  opinion,  who 
thought  that  all  substances  would  become  magnetic  at  a 
sufficiently  low  temperature.  The  following  fact  discovered 
by  Dr.  Hopkinson  seems  to  support  this  opinion:  An  alloy 
of  iron  containing  25  percent,  of  nickel  is  non-magnetic  like 
all  alloys.  But  if  this  alloy  be  cooled  to  slightly  below  o° 
C,  it  is  capable  of  becoming  magnetized  in  a  very  marked 
degree.  It  possesses,  therefore,  a  low  critical  temperature. 
If  the  alloy  be  afterwards  reheated,  it  remains  magnetic  and 
its  susceptibility  increases  up  to  about  525°  C.,  at  this  point 


8O  MAGNETISM. 

the  susceptibility  falls  rapidly  and  becomes  zero  at  580°  C. 
Upon  recooling  the  metal,  it  does  not  reassume  its  suscep- 
tibility until  below  o°  C. 

64.  Ewing's  Addition  to  Weber's  Hypothesis.*— In 

order  to  explain  in  Weber's  hypothesis  of  the  molecular 
constitution  of  magnets,  §  39,  the  coercive  force  and  the 
loss  due  to  hysteresis,  it  has  been  supposed  that  the  ele- 
mentary magnets  (or  magnetic  molecules)  offer  a  resistance 
to  orientation  in  the  nature  of  friction  and  that  it  is  the 
work  spent  in  overcoming  this  friction  which  constitutes  the 
loss  by  hysteresis.  The  existence  of  such  a  passive  resist- 
ance enables  us,  up  to  a  certain  point,  to  account  for  the 
effect  of  vibrations  and  temperature  on  magnets,  but  it  does 
not  at  all  explain  the  changes  in  susceptibility  especially 
shown  in  the  regions  A,  B  and  C  of  the  magnetism-curve, 
Fig.  24. 


FIG.  24. 

Ewing  has  found  experimentally  that  the  observed  phe- 
nomena are  to  be  explained  without  bringing  in  the  supposi- 
tion of  friction,  by  the  simple  effect  of  the  mutual  reactions 
of  the  elementary  magnets.  He  has  reached  this  conclusion 
by  investigating  the  way  in  which  a  system  of  magnetic 
needles  acts,  when  they  are  arranged  regularly  one  next  the 

*  Ewing,  Contributions  to  the  molecitlar  theory  of  induced  magnetism,  Roy, 
Soc.  1890  ;  also  Magnetic  Induction^  etc.,  pp.  287-8  et  seq. 


INDUCED   MAGNETIZATION.  8 1 

other  so  as  to  be  able  to  oscillate  in  the  same  horizontal 
plane  without  touching  each  other.  These  needles  are  sub- 
jected to  a  magnetizing  force  obtained  by  rolling  coils  of 
wire,  carrying  an  electric  current,  around  the  case  enclosing 
them.  When  the  needles  are  left  to  their  own  reactions, 
that  is,  when  the  field  produced  by  the  current  neutralizes 
the  earth's  field,  it  is  observed  that  they  form  more  or  less 
complex  geometrical  combinations  with  each  other  in  stable 
equilibrium.  If  one  of  the  elements  of  this  combination  is 
slightly  altered  from  its  position,  it  immediately  returns  to 
it ;  but  if  the  alteration  of  position  is  considerable,  the  com- 
bination is  not  formed  again,  and  new  combinations  are 
formed  between  the  neighboring  magnets.  If  a  progres- 
sively increasing  directive  force  is  applied  to  such  a  system, 
it  is  seen  that  the  various  combinations  are  at  first  slightly 
deformed  without  being  destroyed.  This,  which  might  be 
termed  an  elastic  deformation  since  it  is  reversible  by  with- 
drawing the  magnetizing  force,  is  comparable  to  the  state  of 
the  molecules  of  a  magnetized  bar  in  the  region  A  of  the 
magnetism-curve,  Fig.  24. 

If  the  current  through  the  coils  is  continuously  increased, 
a  point  is  reached  where  one  of  the  combinations  of  the 
magnets  exceeds  the  limiting  deformation  which  it  can 
stand.  There  is  then  produced  a  sudden  change  in  this 
combination,  and,  by  the  mutual  action,  the  whole  system 
enters  on  a  state  of  unstable  equilibrium,  so  that  a  very 
slight  increase  in  the  directive  force  is  sufficient  to  align  all 
the  magnets  in  a  direction  approaching  to  that  of  the  direc- 
tive force  itself.  This  period  corresponds  to  the  region  B 
of  the  magnetism-curve.  The  observation  of  the  propaga- 
tion of  the  new  groupings  from  one  group  to  the  next  one  is 
eminently  suggestive  as  explaining  the  necessity  of  a  definite 
interval  of  time  for  the  molecules  of  a  magnet  to  assume 


82  MAGNETISM. 

their  positions  of  equilibrium  under  the  action  of  a  magnet- 
izing force. 

When,  after  this,  we  still  continue  to  apply  increasing 
forces,  we  observe  that  the  mutual  reactions  of  the  magnets 
are  more  and  more  overpowered  and  that  they  align  them- 
selves in  a  direction  which  eventually  coincides  with  that  of 
the  acting  field.  This  is  the  state  designated  under  the 
name  of  saturation  and  shown  at  C  in  the  curve. 

Figures  25,  26,  and  27  show  three  successive  states  of  the 
system  of  magnets;  Fig.  25  corresponding  to  the  end  of 
state  A,  Fig.  26  to  the  end  of  state  B,  and  Fig.  27  to  the 
end  of  state  C.  If  we  diminish  the  intensity  of  the  field,  the 
magnets  still  maintain  their  general  orientation,  but  become 
slightly  displaced  by  the  effect  of  their  own  reactions. 
When  the  directing  field  becomes  zero,  the  magnets  remain 
more  or  less  aligned  in  the  direction  of  the  field,  which 
accounts  for  the  residual  magnetism.  But  if  the  directing 
field  changes  sign  and  increases  in  the  opposite  direction, 
we  soon  observe  a  sudden  return  to  the  state  of  unstable 
equilibrium,  and  then  an  orientation  in  the  opposite  direc- 
tion. 

According  to  the  curve  AC  A',  Fig.  18,  annealed  soft  iron 
is  in  a  state  of  unstable  equilibrium  when  the  magnetizing 
force  becomes  zero,  since  the  elbow  of  the  curve  is  on  the 
side  of  the  positive  magnetizing  forces.  With  tempered 
steel,  on  the  contrary,  this  elbow  is  produced  on  the  side  of 
the  negative  abscissae,  which  accounts  for  the  aptitude  of  this 
metal  for  retaining  its  residual  magnetism.  In  fact,  at  the 
moment  when  the  magnetizing  force  becomes  zero,  the  state 
of  the  metal  is  shown  by  a  point  of  the  curve  situated  in  the 
region  corresponding  to  stable  equilibrium. 

If,  instead  of  placing  the  magnets  regularly,  we  arrange 
them  at  varying  distances,  we  observe  that  the  duration  of 
the  state  corresponding  to  unstable  equilibrium  (13)  is  in- 


INDUCED   MAGNETIZATION. 


creased.  This  is  observed  in  cold-hammered  iron,  the 
arrangement  of  whose  molecules  has  been  changed  by 
mechanical  means.  The  various  combinations  of  elements 
which  are  formed  in  such, a"  case  are  more  independent  of 
each  other  than  in  the  homogeneous  metal,  and  one  or 
another  of  them  can  be  modified  without  influencing  the 
adjoining  groupings. 

"    1111  I  1 

~  1 1 1 1 1 1 
I  1 1 1 1  1 1 

-  1 1 1 1  1 1 

-  1 1 1 1 11 

an 


1  ! 

1 1 1 
1 1 1  \ 

lilt 


1 1 

III 

lltl 

1 1 1 1 ' 

nil 


FIG.  25. 


FIG.  26. 


FIG.  27. 


In  steel  and  cast-iron,  where  the  stage  B  is  of  great 
extent,  the  groupings  of  the  molecules  are  influenced  by  the 
presence  of  foreign  bodies. 

According  to  Ewing,  the  heating  due  to  hysteresis  cor- 
responds to  the  oscillations  of  the  magnets  on  passing  from 
one  position  of  stable  equilibrium  to  another. 

Vibrations  diminish  the  stability  of  the  combinations  and 
consequently  facilitate  the  orientation  of  the  magnets  placed 
under  the  action  of  the  field,  as  well  as  their  return  to  the 
neutral  state  when  the  magnetizing  force  has  ceased  to  act. 


84  MAGNETISM. 

A  rise  in  temperature  produces  analogous  effects  when 
the  magnetizing  force  is  feeble ;  we  have  already  seen,  how- 
ever, that  heating  reduces  the  permeability  when  the  mag- 
netizing force  is  intense.  Ewing  explains  this  fact  by 
considering  that  the  molecular  agitation  caused  by  a  rise  in 
temperature  corresponds  to  oscillations  of  the  magnets  about 
their  axes.  When  the  magnets  are  already  oriented,  these 
oscillations  result  in  a  diminution  of  the  mean  external  action 
of  the  system.  Or  we  can  admit,  with  Dr.  Hopkinson,  that 
the  magnetic  moment  of  the  elementary  magnets  decreases 
when  the  temperature  increases. 

Lastly,  the  absence  of  hysteresis,  observed  by  Lord 
Rayleigh  in  the  case  of  very  feeble  variations  in  the  magnet- 
izing force,  can  be  explained  if  we  observe  that  such  varia- 
tions produce  only  feeble  displacements  of  the  elementary 
magnets  about  their  positions  of  equilibrium  ;  such  dis- 
placements are  reversible  without  break  of  equilibrium  and 
consequently  without  the  extensive  movements  which  give 
rise  to  the  development  of  heat. 

The  irreversible  variations,  which  are  made  evident  by 
the  separation  of  the  ascending  and  descending  curves  of 
magnetism,  are  the  only  ones  which  give  rise  to  the  evolu- 
tion of  heat. 

65.  Equilibrium  of  a  Body  in  a  Magnetic  Field.— 
We  have  seen,  §  45,  that  a  shell  free  to  move  in  a  magnetic 
field  moves  so  that  the  flux  entering  by  its  negative  face 
may  be  a  maximum.  This  conclusion  extends  to  any  mag- 
netized body  whatever  which  may  be  considered  as  formed 
by  superposed  shells. 

Thus  in  a  uniform  field  the  axis  of  an  iron  cylinder  of 
elongated  form  aligns  itself  parallel  to  the  lines  of  force  of 
the  field,  in  such  a  way  that  the  flux  of  force  enters  by  the 
induced  south  pole. 


INDUCED    MAGNETIZATION.  85 

It  has  been  shown,  §§53,  55,  that  this  position  corresponds 
to  a  more  intense  magnetization  of  the  metal  than  does  any 
other  position.  In  a  uniform  field  an  isotropic  sphere  is  in 
equilibrium  in  all  positions,  ,while  an  anisotropic  sphere, 
whose  permeability  varies  in  Different  directions,  orients 

•*• 

itself  so  that  the  direction  of  the  field  may  be  parallel  to  the 
axis  of  maximum  permeability. 

The  same  considerations  show  that  in  the  neighborhood 
of  a  magnet  where  the  field  is  variable,  magnetic  bodies 
tend  to  move  towards  the  poles  so  that  the  flux  traversing 
them  may  be  a  maximum. 

These  movements  are  clearly  exhibited  in  a  liquid  placed 
in  a  watch-glass  over  the  poles  of  a  powerful  electro-magnet. 


FIG.  28 

A  solution  of  sulphate  of  iron,  S,  Fig.  28,  will  present  a 
concavity  towards  the  centre. 

Diamagnetic  bodies,  on  the  other  hand,  appear  to  be  re- 
pelled by  the  poles;  thus  a  bar  of  bismuth  assumes  a  posi- 
tion at  right  angles  to  an  electro-magnet  ;  a  solution  of 
bisulphide  of  carbon,  S',  Fig.  28,  is  heaped  up  in  the  middle 
of  the  vessel. 

Becquerel  and  Faraday  have  found  a  simple  explanation 
of  diamagnetic  repulsion,  by  means  of  comparing  it  with  the 
action  of  gravity  upon  bodies  plunged  in  a  liquid  denser 
than  themselves.  These  bodies  are  apparently  repelled  by 
the  earth  ;  in  the  same  way  it  may  be  that  the  action  of 


86  MAGNETISM. 

magnets  on  diamagnetic  bodies  is  simply  due  to  the  fact 
that  they  are  less  magnetic  than  the  air  or  the  medium 
which  surrounds  them.  From  this  point  of  view  there 
would  be,  properly  speaking,  no  diamagnetic  substances, 
but  only  degrees  of  permeability.  This  hypothesis,  corn- 
batted  by  Tyndall,  has  been  recently  confirmed  by  Messrs. 
Parker  and  Duhem. 


HYSTERESIS.  87 


HYSTERESIS    AND    MOCtCULAR    MAGNETIC 
FRICTION.* 

66.  Hysteresis. — Some  materials,  such  as  iron,  nickel,  etc., 
when  exposed  to  the  action  of  a  magnetomotive  force,  that 
is,  when  in  a  magnetic  field,  have  induced  in  them  a  mag- 
netic flux  far  in  excess  of  that  set  up  under  the  same 
conditions  in  air  or  other  materials.  The  former  are  there 
fore  called  magnetic  materials. 

In  air  and  other  non-magnetic  materials  the  magnetic  flux, 
OS,  varies  proportionally  to  the  magnetomotive  force,  £F,  or  to 
the  field  intensity,  5C.  In  magnetic  materials,  such  as  iron, 
the  magnetic  flux  is  proportional  to  the  M.  M.  F.  only  for 
very  low  values  of  the  latter.  With  increasing  M.  M.  F.'s  it 
begins  to  increase  at  a  greater  rate  than  the  M.  M.  F.,  be- 
comes proportional  again  to  it  at  still  higher  values,  and 
for  very  high  M.  M.  F.'s  increases  more  and  more  slowly  until 
ultimately  its  further  increase  with  increase  of  M.  M.  F. 
approaches  a  limit  where  it  is  not  greater  than  in  non- 
magnetic materials ;  or,  in  other  words,  the  difference 
&  —  3C  approaches  a  finite  limit,  called  the  absolute  magnetic 
saturation  of  the  material,-)-  which  in  soft  iron  corresponds 
to  about  (&  =  20,000,  in  nickel  to  <B  =  6000,  in  cobalt  to 
(B  =  13,000,  etc. 

The  curve  indicating  the  variation  of  the  magnetic  flux 
(fc  with  the  M.  M.  F.  or  field  intensity  is  of  a  form  shown 

*  By  Chas.  Proteus  Steinmetz. 

f  "Magnetic  Reluctance,"  by  Kennelly.  Transactions  of  the  American 
Institute  of  Electrical  Engineers,  1891- 


88      HYSTERESIS— MOLECULAR   MAGNETIC  FRICTION. 

in  Fig.  29  by  the  full  line.  This  curve  is  obtained  if  the 
M.  M.  F.  acting  upon  the  iron  is  made  to  gradually  increase 
from  zero  to  a  maximum.  If  now  the  M.  M.  F.  is  gradually 
reduced  again  from  the  maximum  to  zero,  the  correspond- 
ing values  of  magnetic  flux,  (B,  are  not  the  same,  but 
considerably  higher  than  for  increasing  M.  M.  F.,  and  are 


(B 

16000 
15000 
14000 
13000 
12000 
11000 

j-^-i 

—  — 

• 

^— 

p 

^.^ 

^-••"^ 

** 

•^ 

/> 

^ 

x 

f 

' 

'' 

/ 

10000 
9000 
8000 
7000 
6000 
5000 
4000 
3000 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

1 

I 

\ 

\ 

\ 

2000 

1 

1 

.1000 
n 

/ 

x 

FIG.  29.— RISING  AND  DECREASING  MAGNETIC  CHARACTERISTIC. 

shown  in  Fig.  29  in  dotted  line.  Thus  the  magnetic  char- 
acteristic is  different  for  decreasing  and  for  increasing  mag- 
netism ;  or,  in  other  words,  the  magnetic  flux,  (B,  in  mag- 
netic materials,  such  as  iron,  depends  not  only  upon  the 
present  value  of  M.  M.  F.,  but  also  upon  the  previous  values, 
and  thus  lags  behind  the  M.  M.  F. 

This  lag  of   the  magnetic  flux  (B  behind   the  M.  M.  F.  is 


HYSTERESIS. 


89 


practically  independent  of  the  time;  that  is,  independent 
whether  the  change  of  M.  M.  F.  takes  place  rapidly  or  very 
slowly.  It  is  called  hysteresis. 

The  effect  of  hysteresis  upofi  the  magnetic  characteristic 
is  most  pronounced  in  .cyclic  changes  of  flux  as  produced 
by  cyclic  changes  of  M.  M.  F.  Thus  if  the  M'.  M.  F.,  £F,  or  the 


7 


-fUQO 


-KOOO 


+2000 


-4000 


-6000 


-80/0 


^1200 


KB, 
KJC, 


FIG.  30. — HYSTERETIC  LOOP  OR  MAGNETIC  CYCLE. 
field  intensity,  5C,  is  varied  periodically  between  a  maximum 
value  +^1  and  an  opposite  value  —  5C, ,  the  magnetic  flux 
will  vary  between  corresponding  maximum  values  +  (B,  and 
—  (B,  ,  describing  a  loop-shaped  curve  called  the  magnetic 
cycle  or  hysteretic  loop,  as  shown  in  Fig.  30. 


90     HYSTERESIS— MOLECULAR  MAGNETIC  FRICTION. 

It  follows  that  when  the  M.  M.  F.  has  been  reduced  to 
zero,  the  magnetic  flux  has  still  a  considerable  value,  which 
is  called  the  remanent  magnetism,  R  in  Fig.  30. 

The  magnetism  will  reach  zero  only  after  the  M.  M.  F.  has 
been  reversed  and  increased  to  a  considerable  value  £,  in 
opposite  direction.  Thus  the  iron  acts  as  if  an  internal 


FIG.  31. — MAGNETIC  CYCLES  OF  SOFT  SHEET-IRON  OR  SHEET-STEEL. 


M.  M.  F.,  (B,  tends   to    maintain    its    magnetic    flux.     This 
M.  M.  F.  (2.  is  called  the  coercive  force  of  the  iron. 

The  shape  of  the  hysteretic  cycles  varies  with  different 
magnetic  materials,  and  even  with  the  same  magnetic 
material  in  different  physical  conditions,  and  is  different  for 
different  values  of  maximum  magnetic  flux,  &,.  A  number  of 
such  magnetic  cycles  are  shown  in  Figs.  31,  32,  and  33.  Fig. 


H  YS  TERESIS.  9 1 

31  gives  the  magnetic  cycles  of  sheet  iron  or  soft  sheet  steel 
for  different  values  of  magnetic  flux, 

(B  =  2000,     6000,     lopoo,     and     16000.* 

Fig.  32  gives  cast-iron  cycles  for  (B  =  6800  and  (B  =  iO3OO.f 
Fig.  33  gives  cycles  of  tool-steel  at  different  degrees  of  hard- 
ness. H  is  a  sample  hardened  in  water,  O  one  hardened 
in  oil,  and  5  an  annealed  sample,  all  three  of  the  same 


-9O-8O  — 7O-6O  -5O  — 4O  —  SO   - 


0  430 


47O  +8O  •*•  9O 


FIG.  32. — MAGNETIC  CYCLES  OF  CAST-IRON. 

material.!  As  may  be  seen,  the  harder  the  material  the 
lower  and  wider  in  general  is  the  hysteretic  loop  ;  that  is,  the 
lower  the  maximum  and  remanent  flux,  the  higher  the  coer- 


*  "  On  the  Law  of  Hysteresis,"  Part  III,  A.  I.  E.  E.   Transactions,  1894, 
p.  717. 

\  Ibid.,  Part  I,  A.  I.  E.  E.  Transactions,  1892,  p.  40. 
\  Ibid.,  Part  II,  A.  I.  E.  E.   Transactions,  1892,  p.  653. 


92      HYSTERESIS—  MOLECULAR   MAGNETIC  FRICTION. 

cive  force.  In  the  cycles  of  Figs.  31  to  33,  the  abscissae  are 
not  the  field  intensities  3C,  but  the  ampere-turns  per  centi- 
metre length  of  the  magnetic  circuit,  expressed  by 


•p 


103C 


which  is  sometimes  used  as  a  practical  unit  of  M.  M.  F. 

If  an  air-gap  is  introduced  into  the  magnetic  circuit — that 
is,  if  the  magnetic  circuit  is  partly  of  iron  and  partly  of  air, 


FIG>  33. — MAGNETIC  CYCLES  OF  WELDED  STEEL  AT  DIFFERENT  DEGREES 

OF  HARDNESS. 


as  for  instance  in  dynamo  machinery — the  hysteretic  cycle 
changes  to  the  shape  shown  in  Fig.  34,  in  which  the  straight 
dotted  line  represents  the  M.  M.  F.  required  for  the  magnet- 
ization of  the  air-gap,  and  the  hysteretic  loop  has  the  same 
relative  position — that  is,  the  same  horizontal  distance  from 
this  dotted  line — as  it  had  from  the  vertical  line  in  the 
circuit  consisting  entirely  of  iron. 


HYSTERETIC  LOOP. 


93 


As  may  be  seen,  the  effect  of  the  introduction  of  an  air- 
gap  is  to  require  for  the  same  maximum  flux  (Bx  a  much 
greater  M.  M.  F.,  to  reduce  the  remanent  magnetism  very 
greatly,  but  the  coercive  forge  XB  is  not  thereby  affected. 


-50-45- 


48000 


XXK 


-14000 


XX 


+15±20+2543b44 


450 


FIG.  34.—  MAGNETIC  CYCLE  OF  CIRCUIT  CONTAINING  AN  AIR-GAP. 

67.  Hysteretic  Loop.  —  In  the  hysteretic  loop  of  a  mag- 
netic circuit,  with  M.  M.  F.  as  abscissae,  and  magnetic  flux 
as  ordinates,  the  area  of  the  loop  is 

M.  M.  F.  X  magnetic  flux. 
Since  M.  M.  F.  has  the  dimension 


and  magnetic  flux  the  dimension 


where  L  =  length,  M  =  mass,  jT  —  time,  the  area  has  con 
sequently  the  dimension 


that  is,  the  same  as  that  of  energy. 


94     HYSTERESIS—  MOLECULAR   MAGNETIC  FRICTION. 

In  the  hysteretic  loop  with  field  intensity  3C  as  abscissae, 
of  dimension 


and  magnetic  flux  density  (B  as  ordinates,  of  dimension 


the  area  has  the  dimension 


volume 
thus  representing  an  energy  per  unit  volume. 

Let/=:  instantaneous  value  of  M.  M.  F., 
$  =  maximum  "      "  " 

0  =  instantaneous  value  of  magnetism  produced  there- 

by, 

$  =  maximum  value  of  magnetism  produced  thereby. 

If  the  M.  M.  F./is  produced  by  an  alternating  current,  /, 
flowing  through  n  turns,  then 

e  =  -  n^-  =  E.  M.  F. 

Qt 

induced  by  the  magnetism  m,  and 

dw  —  eidt  =  —  mdfi 

=  energy  expended  by  the  change  of  magnetic  conditions. 
Since  ni  —  /, 

dw  =  ~/d0, 
and 

r+s  r-$ 

W=  /d0+ 


MOLECULAR   MAGNETIC  FRICTION.  95 

total  energy  expended  during  the   cyclic   change   of  mag- 
netism ;  but 


/ 


ycl0_|_    /.      /d0  =  area  of  the  hysteretic  loop. 

" 


Therefore,  the  area  of  the  hysteretic  loop,  with  the 
M.  M.  ¥.  in  ampere-turns  as  abscissae,  and  with  the  magnetic 
flux  in  volt-lines  (=  io8  lines,  or  one  hundred  mega- 
webers)  as  ordinates,  is  equal  to  the  energy  expended  by 
hysteresis,  in  coulombs.  With  lines  of  force  as  ordinates, 
and  tens  of  ampere-turns  as  abscissae,  the  area  is  the  hyster- 
etic energy  in  ergs. 

The  area  of  the  hysteretic  loop,  with  field  intensity,  JC,  in 
tens  of  ampere-turns  per  unit  length  of  magnetic  circuit,  as 
abscissae,  and  with  magnetic  flux  density,  (B,  as  ordinates,  is 
equal  to  the  loss  of  energy  by  hysteresis  in  ergs  per  unit 


volume.      With    field    intensity  JC  =  --  as  abscissae,  and 

with  lines  of  force  per  cm.2,  (B,  as  ordinates,  the  energy 
expended  by  hysteresis  during  a  complete  cycle  of  magnet- 
ization is  =  47T  X  area  of  hysteretic  loop. 

Hysteresis  thus  represents  an  expenditure  of  energy  by 
the  M.  M.  F.  and  is  measured  by  the  area  of  the  hysteretic 
loop  or  magnetic  cycle. 

68.  Molecular  Magnetic  Friction.  —  If  by  an  alternating 
M.  M.  F.  an  alternating  magnetic  flux  is  produced  in  iron 
or  other  magnetic  material,  a  loss  of  energy  takes  place 
in  the  iron  by  a  kind  of  frictional  resistance  of  the  molecules 
against  the  change  of  their  magnetic  condition.  This  phe- 
nomenon is  called  molecular  magnetic  friction.  Therefore, 
to  alternate  a  magnetic  flux,  energy  has  to  be  expended 
upon  the  iron. 


g        HYSTERESIS— MOLECULAR  MAGNETIC  FRICTION. 

If  the  alternation  of  the  magnetic  flux  is  produced  by 
an  alternating  current,  and  the  condition  is  such  that  no 
energy  is  expended  upon  the  magnetic  circuit  by  any  other 
source,  nor  external  work  done  by  the  magnetic  circuit,  the 
energy  consumed  by  molecular  magnetic  friction  has  to 
be  supplied  by  the  alternating  current.  Consequently,  the 
magnetic  flux  cannot  follow  the  M.  M.  F.,  but  must  lag 
behind  it  so  far  that  the  hysteretic  curve  of  magnetic  flux 
and  M.  M.  F.  represents  the  energy  expended  by  molecular 
magnetic  friction. 

It  follows  that  in  an  alternating  magnetic  circuit  which 
neither  produces  external  work  nor  receives  energy  from 
another  source  than  the  alternating  M.  M.  F.,  the  energy  con- 
sumed by  molecular  magnetic  friction  is  equal  to  the  energy 
expended  by  magnetic  hysteresis. 

If,  however,  external  work  is  done  by  the  magnetic  circuit, 
or  work  expended  upon  it  by  an  external  force,  the  identity 
between  the  energy  of  molecular  magnetic  friction  and  the 
energy  of  magnetic,  hysteresis  no  longer  exists.  Thus,  if 
the  magnetic  circuit  is  vibrated  mechanically  during  the 
cycle  of  magnetization,  the  hysteretic  loop  collapses  more  or 
less  completely,  and  the  rising  and  decreasing  magnetic 
characteristics  coincide  ;  the  energy  consumed  by  molecular 
magnetic  friction  being  supplied  in  this  case  from  the  mechan- 
ical source  vibrating  the  magnetic  circuit.  Conversely,  if 
mechanical  work  is  done  by  the  magnetic  circuit,  as,  for  in- 
stance, if  the  magnetic  circuit  consists  of  iron  filings  or 
loose  laminations  which  can  vibrate  and  rearrange  them- 
selves, the  hysteretic  loop  is  greatly  extended  and  represents 
not  only  the  energy  consumed  by  molecular  magnetic  fric- 
tion, but  also  the  mechanical  work  done.* 

*  For  proof  and  discussion  of  the  distinction  between  hysteresis  and  mo- 
lecular magnetic  friction  see:  "On  the  Law  of  Hysteresis,"  Part  II,  Chap. 
V,  A.  I.  E.  E.  Transactions,  1892,  p.  711,  and  "On  the  Law  of  Hysteresis," 
Part  III,  Chap.  II,  A.  I.  E.  E.  Transactions,  1894,  p.  706. 


DETERMINATION   OF   VALUES.  97 

It  follows  that  in  determining  the  energy  loss  bymolecular 
magnetic  friction  from  the  hysteretic  loop  of  the  material, 
care  must  be  taken  that  neither  external  work  is  done  nor 
absorbed  by  the  magnetic  circuit  while  the  hysteretic  loop 

is  being  determined. 

•R 
-*• 

69.  Determination  of  Hysteresis  and  Molecular  Mag- 
netic Friction. — The  different  methods  of  determining  the 
value  of  hysteresis  and  molecular  friction  are  as  follows: 

(a)  Ballistic  Method. — A  magnetic  circuit  is  built  up  of  the 
iron  to  be  tested,  a  magnetizing   coil  wound  around  it  as 
uniformly  as  possible,  and  a  second  or  exploring  coil    em- 
ployed, connected  to  a  ballistic  galvanometer.    The  current 
in   the  magnetizing  coil  is  varied    step   by    step,    and    the 
time  integral  of  E.  M.  F.  induced  in  the  exploring  coil  by 
the  variation  of  current,   and  consequently  the  change   in 
magnetic  flux,  is  observed  by  means  of  the  ballistic  galvanom- 
eter.    In  this  way  a  complete  cycle  of  magnetism  is  plotted 
and    from    its    area  the    loss   of  energy  determined.     This 
method    does    not    give    the    energy  of    molecular    friction 
directly,  but  gives  the  energy  expended  by  hysteresis.     It 
can  be  used  for  the  determination  of  the  saturation  curve 
also,  and  is  suitable  for  the  investigation  of  solid  materials, 
as  well  as  of  laminations,  etc.     In  determining  the  saturation 
curve  by  this  method,  it  is  desirable  to  dissipate  the  remanent 
magnetism  previous  to  the  test,  by  applying  a  strong  alternat- 
ing current  through  the  magnetizing  coil,  and  gradually  re- 
ducing this  current  to  zero.       In  determining  the  hysteretic 
cycle,  a  greater  number  of  cycles  between  the  same  maxi- 
mum values  should  be  described  before  taking  readings,  so 
as  to  make  the  cycle  symmetrical  and  independent  of  rema- 
nent magnetism  due  to  the  previous  history  of  the  iron. 

(b)  Alternate  Current  Method. — The   energy  expended  by 
magnetic  hysteresis  can  be  determined  directly  by  sending 


98      HYSTERESIS— MOLECULAR   MAGNETIC  FRICTION. 

an  alternating  current  through  a  magnetizing  coil  surround 
ing  the  magnetic  circuit,  and  taking  readings  of  an  ammeter, 
a  voltmeter,  and  a  wattmeter.  The  M.  M.  F.  is  determined 
from  the  ammeter  reading,  the  magnetic  flux  by  calcula- 
tion from  the  voltmeter  reading,  number  of  turns,  fre- 
quency and  shape  of  the  magnetic  circuit,  and  the  energy 
loss  by  hysteresis  from  wattmeter  readings. 

This  method  is  applicable  only  to  laminated  material,  and 
measures  not  only  the  energy  expended  by  hysteresis  but 
also  the  energy  loss  by  eddy  or  Foucault  currents.  Conse- 
quently, either  the  material  has  to  be  subdivided  so  as  to 
make  the  latter  negligible,  or  hysteresis  and  eddy  currents 
have  to  be  separated  from  each  other  afterwards. 

Since  the  maximum  flux  and  maximum  M.  M.  F.  in  this 
method  are  obtained  by  calculation  from  the  effective 
values  read  on  the  instruments,  the  wave  of  E.  M.  F.  should 
be  as  near  as  possible  a  sine  wave.  Owing  to  the  distortion 
of  the  current  wave  by  hysteresis,  the  maximum  value  of 
current,  even  with  a  sine  wave  of  E.  M.  F.,  is  not  through- 
out the  whole  range  equal  to  ^/ 2  times  the  effective  value. 
Calculating  the  M.  M.  F.  under  the  assumption  of  a  sine 
wave  of  current,  therefore,  gives  a  magnetic  characteristic, 
which,  while  practically  coinciding  with  the  true  magnetic 
characteristic  within  a  range  up  to  (B  =  10,000  or  14,000,  yet 
differs  greatly  therefrom  beyond  this  range,  showing  appar- 
ently very  high  values  of  flux.  In  Fig.  35  is  shown  the 
true  magnetic  characteristic  in  full  line,  the  magnetic  char- 
acteristic as  determined  from  an  alternating  current  test 
in  dotted  line,  and  the  loss  of  energy  by  hysteresis  in  ergs 
per  cm3  and  cycle  in  the  full  line  of  single  curvature.* 

Instead  of  connecting  the  voltmeter  and  the  potential 
coil  of  the  wattmeter  across  the  magnetizing  coil  of  the 

*  "On  the  Law  of  Hysteresis,"  Part  III,  A    I.  E-  E.  Transactions,  1894, 
p.  722. 


DETERMINATION  OF   VALUES. 


99 


magnetic   circuit,  it   is   preferable  to  connect   it   across  a 
second  or  exploring  coil  wound  uniformly  over  the  magnetic 


,cr 

4 

II 

nf\ 

... 

/ 

1     / 

1  I 

I 

19 

/ 

1 
1 
1 

1 

1 
1 

Ifi 

/    - 

1 

/ 

/ 

/ 

•10  . 

/ 

/ 

1 

A 

// 

U 

/ 

1 

1 
1 

/ 

/, 

1 

1U 

/ 

//' 

0 

/ 

/ 

/, 

/ 

/ 

V' 

/ 

// 
// 
// 

/ 

^ 

4 

/ 

^ 

^ 

/ 

rr-s5 

^ 

^ 

^^ 

"^ 

^^ 

1          > 

^ 

x*1 

/ 

/ 

^ 

^~ 

(B=l,000  2,000  3,000  4,000  5,000  6,000  7,000  8,000   9,000  10,00011,00012,00013,60014,00015,00016,00017,000 

FIG.  35.—  MAGNETIC  CHARACTERISTIC  FROM  BALLISTIC  AND  ALTERNATING 
CURRENT  TESTS,  AND  MOLECULAR  MAGNETIC  FRICTION  CURVE. 

circuit,  as  shown  diagramatically  in  Fig.  36,  and  thereby 
eliminate  the  error  due  to  the  drop  of  voltage  and  loss  of 
energy  in  the  resistance  of  the  magnetizing  coil. 


100    HYSTERESIS— MOLECULAR  MAGNETIC  FRICTION. 

(c)  Power  and  Torque  Tests. — The  loss  of  energy  by  mo- 
lecular magnetic  friction  and  eddy  currents  may  be  deter- 
mined by  moving  the  material  to  be  tested  in  a  uniform 
magnetic  field,  and  measuring  the  power  required  therefor. 

This  method  is  commonly  used  in  determining  the  hyster- 
etic  loss  in  the  armatures  of  dynamo  machinery.  In  this 
case  the  armature  is  turned,  first  in  an  unexcited  magnetic 
field,  and  then  in  a  magnetic  field  of  various  degrees  of 


FIG.  36.— INSTRUMENT  CONNECTIONS  FOR  ALTERING  CURRENT  HYSTERESIS 

TESTS. 

excitation.  The  difference  in  work  consumed  in  these  cases 
is  the  energy  expended  in  molecular  magnetic  friction  and 
eddy  currents  in  the  armature. 

A  similar  method  suitable  for  testing  of  iron,  is  the 
following  : 

A  uniform  magnetic  field  is  set  in  rotation,  and  in  the 
centre  of  the  field  the  iron  to  be  tested  is  suspended  by  a 
torsion  spring.  Owing  to  the  hysteretic  loss  in  the  iron,  the 


LOSS   OF  ENERGY.  IOI 

rotating  field  tends  to  turn  the  iron  in  its  direction  of  rota- 
tion, and  the  torsion  given  to  the  spring  to  counterbalance 
this  torque  is  directly  proportional  to  the  hysteretic  loss  per 
cycle  in  the  test-piece.  The  magnetic  flux  is  measured  by 
the  E.  M.  F.  induced  in  an  coloring  coil  surrounding  the 
test-piece.*  This  method  has  the  advantage  that  the  torque 
exerted  upon  the  iron  by  molecular  magnetic  friction  is 
independent  of  the  speed  of  the  rotating  field. 

70.  Loss  of  Energy. — The  hysteretic  cycle  has  been 
found  identically  the  same  from  very  slow  alternations,  up  to 
the  highest  frequencies  reached  by  dynamo-electric  machin- 
ery— beyond  200  cycles  per  second.  It  can,  therefore,  be 
said  that  hysteresis  and  the  loss  of  energy  by  molecular 
magnetic  friction  per  cycle  are  independent  of  the  frequency, 
and  consequently  the  loss  of  power  is  proportional  to  the 
frequency,  that  Is,  to  the  number  of  cycles. 

No  difference  between  the  value  of  hysteretic  energy  and 
loss  by  moleculai  magnetic  friction  has  been  found  between 
rotating  and  reversing  fields,  and  therefore  it  is  permissible 
to  use  the  values  found  in  alternating  fields  for  losses  in 
rotating  fields,  and  conversely. 

The  hysteretic  loop  increases  with  increasing  maximum 
magnetic  flux  density  (B  ;  that  is,  the  energy  expended  by 
hysteresis  per  cycle,  and  the  energy  absorbed  by  molecular 
magnetic  friction,  both  increase  with  the  magnetic  flux. 
Plotting  the  loss  of  energy  as  a  function  of  the  magnetic 
flux,  the  curve  thus  obtained  rises  uniformly,  and  does 
not  show  any  marked  features  at  the  critical  points  of  the 
magnetic  saturation  curve,  following  the  same  law  at  satu- 
ration and  below  saturation. 

Approximately,  the  loss  of  energy  by  molecular  magnetic 

*  Holden,   The  Electrical  World,  June  15,  1895. 


102     HYSTERESIS—  MOLECULAR   MAGNETIC  FRICTION. 
friction  can  be  expressed  by  the  empirical  formula,* 


where  WH  =  loss  of  energy  in  ergs  per  cycle  and  cma,  ±(B  = 
maximum  values  of  magnetic  flux  between  which  the  mag- 
netic cycle  is  performed,  and  rj  is  the  coefficient  of  molecular 
magnetic  friction. 

The  same  empirical  function  of  the  1.6  power  holding  for 
reversals  of   magnetism,  holds  also  for  cyclic   changes   of 


oc 

1S.CCO 

/ 

\ 

/ 

/ 

10,000 

8000 

I 

/ 

X 

/ 

2 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

sooo 

X 

/ 

^ 

^> 

~Z? 



.—  — 

—  " 

,s 

>,=  2000   MOO   6000  8000   10,000  12,000  14,000  16,000  I8.0UO 

FIG.  37. — CURVE  OF  HYSTERESIS  OF  SOFT  IRON  WIRE. 

flux  between  any  two  limits  (B,  and  &3 ,  whether  these 
two  limits  be  of  the  same  direction  and  same  sign  or  of 
opposite  direction.  Thus,  in  general  form  the  empirical  law 
of  molecular  magnetic  friction  can  be  expressed  by  the 
formula : 


*"On  the   Law  of  Hysteresis,"  A.  I.  E.  E.   Transactions^  1892,  p.  3  and 
p.  621;  1894,  p.  702. 


LOSS   OF  ENERGY . 


103 


This  equation  can  be  considered  as  empirical  only,  and,  in 
fact,  does  not  hold  for  extremely  low  values  of  magnetic  flux  ; 
it  is,  however,  correct  with  sufficient  approximation  through 
a  wide  range,  as  shown  in  f  igs.  37  and  38,*  which  give  the 
observed  values  of  hysteretic^Joss  in  a  sample  of  soft-iron 
wire,  and  in  a  sample  of  annealed  steel  wire,  as  taken  from 
Ewing's  tests.  The  curve  of  1.6  power  is  shown  in  full 


7 

UV 

/ 

1 

/ 

Ma 
ed 

jnetic  Cycles  of 
Pianoforte  Stee 
;Ewing,p.109)  . 

/ 

60,000 

Ar 

nee 

I  W 

re. 

/ 

/ 

7 

60,000 

1 

'/ 

/ 

/ 

7 

/ 

20,000 
IQjOOO 

/ 

7 

/ 

/ 

' 

*£ 

^• 

•"' 

^ 

"/? 

^ 

/ 

/_ 

** 

X 

1 1 1  I 

FIG.  38. — CURVE  OF  HYSTERESIS  OF  ANNEALED  PIANOFORTE  STEEL  WIRE. 

line  in  these  figures,  with  the  observed  values  marked  by 
crosses. 

The  dotted  line  represents  the  magnetic  characteristic. 

In  reality,  the  value  (ft  in  the  formula  of  hysteresis  should 
probably  not  be  the  total  magnetic  fiux,  but  the  "  metallic 
magnetic  flux,"  or  &  —  5C,  which  reaches  a  finite  value  of 


*  "On  the  Law  of  Hysteresis,"  Part  II,  A.  I.  E.  E.   Transactions,    1892, 
pp.  673,  674 


IO4     HYSTERESIS— MOLECULAR   MAGNETIC  FRICTION. 

absolute  saturation,  while  (B  increases  indefinitely  with  in- 
creasing OC.* 

71.  Coefficient  of  Hysteresis.— The  coefficient  of  mo- 
lecular magnetic  friction  varies  very  greatly  with  different 
materials  and  even  with  different  conditions  of  the  same 
material. 

In  iron,  the  loss  by  molecular  magnetic  friction  seems  to 
depend  comparatively  little  upon  the  chemical  constitution. 
In  general,  the  purer  the  iron  is,  the  lower  the  co- 
efficient ??,  and  the  less  the  loss  by  molecular  friction. 
Frequently,  however,  very  pure  samples  of  iron  show  com- 
paratively high  values  of  77,  while  impure  samples  show  re- 
markably low  hysteretic  losses.  A  large  percentage  of 
carbon,  silicon,  and  phosphorus  seems  to  be  objectionable, 
while  manganese  in  small  percentages  appears  comparatively 
harmless.  Even  here,  however,  the  effect  of  impurities 
seems  to  be  indirect,  and  due  to  changes  in  the  physical 
constitution  of  the  material. 

Of  the  greatest  importance  regarding  hysteretic  loss  is  the 
physical  condition  of  the  material,  one  and  the  same  material 
in  a  hardened  state  occasionally  showing  a  hysteretic  loss 
many  times  larger  than  when  in  annealed  state.  Annealing 
is  always  found  to  reduce  the  loss  by  molecular  friction 
more  or  less,  while  hardening  increases  it.  With  tool  steel, 
for  instance,  the  coefficient  of  hysteresis  has  been  varied  from 
14.5  to  75  by  annealing  and  hardening,  respectively.  (See 

Fig.  330 

Break  of  continuity  of  the  material  usually  greatly  in- 
creases the  loss  by  molecular  magnetic  friction.  Thus,  gray 
cast-iron,  even  when  very  soft,  shows  a  comparatively  high 

*  "  On  the  Law  of  Hysteresis,"  Part  II,  A.  I.  E.  E.  Transactions,  1892, 
Chap.  Ill,  pp.  678  seq. 


COEFFICIENT   OF  HYSTERESIS.  1 05 

coefficient,  77,  due  to  the  interposition  of  graphite  in  the 
metallic  structure. 

The  temperature  affects  the  loss  of  energy  by  molecular 
magnetic  friction  very  little,  within  the  range  of  atmospheric 
temperatures,  the  hysteretic^loss  decreasing  considerably 
only  at  high  temperatures.  If  iron  is  heated  and  then 
cooled,  the  loss  by  molecular  friction  is  much  smaller  when 
hot,  and  is  not  increased  again  to  the  same  value  as  before 
when  cooled  ;  a  part  of  the  decrease  due  to  the  heating  be- 
comes permanent,  probably  due  to  a  change  of  physical 
condition.  This  is  especially  noticeable  with  steel.  After 
repeated  heating  and  cooling,  the  variation  becomes  reversi- 
ble and  the  hysteretic  loss  decreases  with  increasing  temper- 
ature and  increases  again  to  the  same  value  with  decreasing 
temperature,  approximately  as  a  linear  function  of  the 
temperature.* 

Mechanical  action  affects  the  energy  expended  by  hystere- 
sis very  greatly,  and  causes  the  hysteretic  loop  to  collapse 
more  or  less,  but  apparently  does  not  affect  the  loss  of 
energy  by  molecular  magnetic  friction ;  a  distinction  thus 
exists  between  hysteretic  loss  and  molecular  magnetic 
friction  loss. 

Very  long-continued  exposure  of  iron  in  alternating  mag- 
netic fields  seems  in  some  cases  to  increase  the  loss  by 
molecular  magnetic  friction  through  what  has  been  called 
ageing  of  the  iron,  which  has  been  observed  especially  in 
iron  with  very  low  coefficient  77.  Other  careful  tests,  how- 
ever, have  not  shown  any  trace  of  an  increase  of  hysteretic 
loss  during  continuous  use  in  alternating  fields,  and  the  ob- 
served increase  thus  appears  to  be  due  to  secondary  causes, 
probably  to  a  continued  heating  in  the  alternating  field 
beyond  a  critical  point  of  the  iron,  and  a  change  of  the 

*  W.  Kunz,  Electrotechnische  Zeitschrift,  Berlin,  April  5,  1894. 


106     HYSTERESIS— MOLECULAR   MAGNETIC  FRICTION. 

physical  condition  caused  thereby.  An  effect  similar  to 
that  observed  in  the  crystallization  of  wrought-iron  in 
bridges,  etc.,  under  vibrating  stresses,  may  also  account  for 
the  so-called  ageing. 

The  following  values  of  the  coefficient  of  molecular  mag- 
netic friction  have  been  observed  in  different  materials.* 

COEFFICIENT    OF    HYSTERESIS,    AND    ABSOLUTE    MAGNETIC 
SATURATION. 


Coefficient  in 
Milliunits. 

Average 

Absolute 
Magnetic  Satu- 
ration, 

(B  -  3C  =  4ir/ 

in  Kilolines. 

Soft  sheet-iron  'and  ^heet-steel   

I    2J.-5    ^ 

2  ^—3  ^ 

17   20 

Cast-iron  

II  .3—16.2 

iq 

IO—  II 

Cast-steel  of  low  permeability             . 

12 

j  j 

"         of  high           "             soft  .... 

332—  Q 

6 

12    IQ  ^ 

"         of     "                           hard  .... 

28 

18.5 

Welded  steel  annealed     

14.   ^ 

17.4. 

"         "      oil  hardened  

27 

16.7 

"         "      very  hard 

7c 

8  ^ 

Manganese  steel,  annealed,  4.7$  Mn.  .  . 

41 

8.74$  Mn.. 

82 

"               "     oil  hardened,  4.7$  Mn 

67 

Chrome-steel,  annealed    I  2%  Cr  

16 

"      oil  hardened,  1.2%  Cr.  .  .  . 

44 

Wolfram  steel   annealed    4  6$  W^o   .  .  .  . 

14. 

"       oil  hardened,  3.44$  Wo. 

48 

Magnetite  

20.4-23.5 



4-7 

Nickel  wire    soft                                  . 

12  2—15  6 

I  3 

"           "      hardened          ...... 

18  * 

1  J 

•v 

Cobalt    cast  

II    Q 

Amalgam  of  iron,  1  1%  Fe   

231 

•V 

While  the  loss  of  energy  by  hysteresis  and  by  molecular 
magnetic  friction  is  independent  of  the  frequency  within  a 
very  wide  range,  for  extremely  slow  variations  of  M.  M.  F. 
a  time  lag  exists,  especially  for  very  low  M.  M.  F.'s.  That 


*  "  On  the  Law  of  Hysteresis,"  Part  II,  A.  I.  E.  E.  Transactions,  1892,  p. 
680;  Part  III,  1894,  p.  705. 


EDDY   CURRENTS.  IO/ 

is,  after  the  application  of  the  M.  M.  F.  the  magnetism  rises 
quickly  to  a  certain  value  and  then  keeps  on  rising  slowly 
for  seconds  and  even  minutes  until  a  final  second  value  is 
reached.  In  alternating  current  machinery  this  phenomenon 
of  magnetic  sluggishness  is  o£<  no  importance. 

jp{ 

72.  Eddy  Currents. — In  magnetic  materials  exposed  to 
an  alternating  magnetic  field,  besides  the  loss  of  energy 
by  magnetic  hysteresis  or  molecular  magnetic  friction,  a 
further  loss  of  energy  takes  place  by  eddy  currents  or 
Foucault  currents.  The  eddy  currents  are  not  a  magnetic 
phenomenon  like  hysteresis,  but  a  purely  electrical  phe- 
nomenon. They  are  secondary  currents  induced  in  the 
iron  by  the  alternating  magnetic  field.  Thus,  while  the 
loss  of  energy  by  molecular  magnetic  friction  is  entirely 
independent  of  the  shape  of  the  magnetic  circuit,  and  is 
the  same  whether  the  material  is  solid  or  subdivided, 
the  loss  of  energy  by  eddy  currents  depends  entirely  upon 
the  shape  of  the  iron,  and  though  often  excessive  in 
solid  iron,  may  be  made  small  or  negligible  by  thorough 
subdivision. 

Eddy  currents  are  true  secondary  or  induced  currents, 
and  therefore  their  E.  M.  F.  is  proportional  to  the  magnetic 
flux  (B  and  the  frequency  N\  consequently  the  loss  of 
energy  by  eddy  currents  is  proportional  to  (Ba  and  IV*.  The 
loss  of  energy  per  cycle  is  proportional  to  the  square  of  the 
magnetic  flux,  (B,  and  directly  proportional  to  the  fre- 
quency N,  while  the  loss  of  energy  of  molecular  magnetic 
friction  is  proportional  to  the  1. 6th  power  of  the  magnetic 
induction  (B  and  independent  of  the  frequency  N. 

The  loss  of  energy  by  eddy  currents  per  cm*  and  cycle 
may,  therefore,  be  represented  by  the  formula 

We  = 


108     HYSTERESIS— MOLECULAR   MAGNETIC  FRICTION. 

and  the  total  loss  of  energy  in  the  iron  by  molecular  mag- 
netic friction  and  eddy  currents  by  the  formula 

W=  WH  +  We  =  r}&'*  +  eN&\ 

where  ?;=  coefficient  of  hysteresis  ; 

e  =  coefficient  of  eddy  currents  ; 
N  =  frequency ; 

OS  =  maximum  magnetic  flux  density  ; 
W  =  WH  +  We  =  total  loss  of   energy,  in    ergs   per 
cm8  and  cycle.* 

This  formula  of  the  total  loss  of  energy  allows  the  separa- 
tion of  the  loss  by  molecular  magnetic  friction  from  the  loss 
by  eddy  currents,  by  means  of  two  experimental  observa- 
tions. This  can  be  done  either  by  making  two  observations 
for  the  same  flux  (B  and  different  frequencies  TV,  and  N9 , 
or  by  making  two  observations  at  the  same  frequency  N 
for  different  magnetic  fluxes  (B,  and  (Ba.  In  either  case  we 
get  two  equations  with  two  unknown  quantities  rj  and  e. 
The  latter  method,  with  different  magnetic  fluxes  and  the 
same  frequency,  is  very  convenient,  but  depends  upon 
the  shape  of  the  curve  of  molecular  magnetic  friction — that 
is,  upon  the  empiric  law  of  the  i.6th  power — while  the  former 
method  of  observation  at  different  frequencies  is  independ- 
ent thereof. 

In  general,  it  is  preferable  to  make  a  considerable  number 
of  observations,  plot  them  in  a  curve,  and  take  two  points 
from  the  curve  as  far  apart  from  each  other  as  is  possible 
without  introducing  the  increased  error  of  observation  at 
the  limits  of  the  range  of  test. 

*  "On  the  Law  of  Hysteresis,"  Part  I,  A.  I.  E.  E.  Transactions,  1892,  p. 
22.  For  calculation  of  the  coefficient  of  eddy  currents  for  laminations  and 
wire,  see  "On  the  Law  of  Hysteresis,"  Part  III,  A.  I.  E.  E.  Transactions, 
1894,  pp.  734-738. 


EDDY   CURRENTS.  1 09 

73.  Effect  of  Molecular  Magnetic  Friction  and  Eddy 
Currents. — Eddy  currents  can  be  eliminated  by  thorough 
subdivision  of  the  magnetic  material,  and  therefore  no 
excuse  exists  for  their  presence  in  any  first-class  electrical 
machinery. 

Molecular  magnetic  friction  represents  a  loss  of  energy 
which  cannot  be  avoided  by  subdivision,  but  can  be  greatly 
reduced  by  the  choice  of  the  best  available  iron.  This  loss 
of  energy  is  present  wherever  magnetic  fields  are  alternating, 
and  is,  therefore,  found  in  the  armatures  of  continuous  cur- 
rent dynamos  and  motors ;  to  a  larger  extent  still  in  the 
armatures  of  alternators,  due  to  the  higher  frequency  at 
which  they  usually  operate ;  and  in  transformers  and  induc- 
tion motors. 

In  continuous  current  machinery  the  loss  by  molecular 
magnetic  friction  is  usually  only  a  comparatively  small 
part  of  the  total  loss.  In  alternators  the  molecular 
magnetic  friction  loss  is  one  of  the  largest  losses,  and 
frequently  even  larger  than  all  the  other  losses  combined. 
It  is  therefore  of  the  greatest  importance  in  alternating 
practice  to  use  the  best  possible  iron.  It  is  obvious  that 
neglect  of  the  molecular  magnetic  friction  in  the  determina- 
tion of  the  efficiency  of  alternators  causes  the  efficiencies  to 
appear  very  much  higher  than  they  really  are.  Consequently 
the  so-called  electrical  efficiency  of  alternators  is  of  no  value 
whatever,  and  in  discussing  the  efficiency  of  alternators, 
care  has  to  be  taken  to  include  the  loss  of  molecular  mag- 
netic friction  ;  this  has  not  always  been  done,  and  as  a  result 
efficiencies  have  been,  and  still  are  being,  quoted  in  excess 
of  any  reasonable  value.  When  speaking  of  alternators, 
obviously  alternate  current  generators  as  well  as  synchron- 
ous motors  are  understood. 

In  transformers  the  loss  from  molecular  magnetic  friction 
is  usually  the  largest  of  the  several  losses,  especially  in 


HO    HYSTERESIS— MOLECULAR   MAGNETIC  FRICTION. 

larger  sizes  of  transformers,  and  is  the  more  undesirable  as 
it  occurs  equally  at  all  loads  and  at  no  load,  while  the  resist- 
ance loss  in  the  electric  conductors  is  significant  only  with 
a  considerable  load.  Thus  the  all-day  efficiency  of  a  trans- 
former, or  the  ratio  of  the  total  amount  of  energy  put  into 
the  primary  of  the  transformer  to  the  useful  energy  taken 
out  at  its  secondary,  depends  almost  exclusively  upon  the 
molecular  magnetic  friction  loss.  The  same  applies  to 
induction  motors,  in  so  far  as  they  are  transformers. 

74.  Equivalent  Sine  Curves. — A  further  effect  of  mo- 
lecular  magnetic  friction  is  its  action  on  the  shape  of  the 
current  wave.  If  a  sine  wave  of  E.  M.  F.  is  impressed  upon 
a  magnetic  circuit,  a  current  will  be  produced  in  the  circuit 
which  is  not  a  sine  wave,  but  differs  widely  therefrom,  being 
distorted  by  hysteresis. 

With  a  negligible  internal  resistance,  the  sine  wave  of  im- 
pressed E.  M.  F.  implies  a  sine  wave  of  counter  E.  M.  F.  or 
induced  E.  M.  F. — that  is,  a  sine  wave  of  magnetism,  or  of 
magnetic  flux,  (B.  Owing  to  the  discrepancy  between 
M.  M.  F.  and  magnetic  flux  as  represented  in  the  hysteretic 
cycle,  the  M.  M.  F.,  and  therefore  the  current  required  to 
produce  the  sine  wave  of  magnetic  flux,  is  not  a  sine  wave. 
It  is  determined  by  taking  from  the  sine  wave  of  mag- 
netic flux  (B  the  instantaneous  values  of  (B,  and  from 
the  hysteretic  curve  the  values  of  M.  M.  F.,  and  therefore 
of  current  corresponding  to  the  instantaneous  values  of 
magnetic  flux  (B,  and  plotting  these  currents  as  func- 
tions of  the  time  in  the  same  way  as  magnetic  fluxes 
are  plotted  as  sine  waves.  In  this  way  different  current 
waves  are  produced  for  different  maximum  magnetic  fluxes. 
A  number  of  such  distorted  current  waves  are  shown 
in  Figs.  39  to  42,  corresponding  to  values  of  maximum  flux 


EQUIVALENT  SINE    CURVES. 


Ill 


(B  =  2000,    600O,    IO,OOO,    16,000 

and  of  M.  M.  F.'s, 

$  =      1.8,        2.8,  4.3,          20.0, 

1 

and  also  to  the  hysteretic  cycles :-ghown  in  Fig.  31.     As  may 
be  seen,  all  these  curves  are  bulged  out  at  the  ascending, 


^2000 


=  =  1.8 


v/ 


=  6000 


r2.8 


^ 


Bradley  $  Poatet,  Engr't,  ff.f. 

FIGS.  39  AND  40. — DISTORTION  OF  CURRENT  WAVE  BY  HYSTERESIS. 

and  hollowed  in  at  the  descending  side.  With  increasing 
values  of  maximum  magnetic  flux,  the  maximum  point  of  the 
current  wave  becomes  pointed,  as  in  Fig.  41,  and  ultimately 
a  sharp  high  peak  is  formed,  as  in  Fig.  42,  due  to  magnetic 


112     H  YS  TERESIS—MOLECULA  R   MA  GNE  TIC  FRICTION. 


saturation.*  Such  distorted  current  waves  can,  like  most 
distorted  alternating  waves,  be  replaced  for  all  practical 
purposes  by  equivalent  sine  waves  ;  that  is,  sine  waves  equal 
in  effective  intensity  and  in  energy  to  the  distorted  wave. 
These  equivalent  sine  waves  of  exciting  current  are  shown 


\\ 


-Cj 


16000 


20 


\ 


13 


\ 


\ 


\J 


Bradley  &  Poatee,  Engr's,  X.  Y. 

FIGS.  41  AND  42.  —  DISTORTION  OF  CURRENT  WAVE  BY    HYSTERESIS. 

in  Figs.  39  to  42  ;  the  remainder,  or  the  difference  between 
the  true  distorted  current  wave  and  the  equivalent  sine 
wave,  is  shown  also,  and  consists,  as  seen,  essentially  of  a 
term  of  triple  frequency.  This  means  that  hysteresis  intro- 


On  the  Law  of  Hysteresis,"  Part  III,  A.  I.  E.   E.   7*ransactions,  1894, 


p.  719- 


EQ  UI  VA  L  EN  T  SINE    CUR  VES.  1  1  3 

duces  higher  harmonics  in  the  current  wave,  amongst  which 
the  triple  harmonic  is  especially  pronounced. 

The  equivalent  sine  wave  of  exciting  current  is  not  in 
phase  with  the  wave  of  magnetfem,  but  leads  it  by  an  angle 
a,  which  is  called  the  angle  of  feysteretic  advance  of  phase. 
In  consequence,  the  exciting  current  can  be  resolved  into 
two  components  ;  one, 

/  cos  a, 

in  phase  with  the  magnetism  —  that  is,  in  quadrature  with 
the  induced  E.  M.  F.,  or  wattless,  which  is  called  the  magnet- 
izing current  ;  and  into  another  component, 

/sin  a 

in  quadrature  with  the  magnetism  —  that  is,  in  phase  with 
the  induced  E.  M.  F.,  representing  consumption  of  energy 
and  called  the  magnetic  energy  current. 

The  ratio  of  the  magnetic  energy  current  to  the  total 
exciting  current  is  the  so-called  "  power  "  factor  of  the 
electromagnetic  circuit,  whose  value  is 


sin  a  = 


that  is,  it  depends  upon  the  coefficient  of  hysteresis  77,  the 
magnetic  permeability  //  and  the  maximum  magnetic  flux 
(&,  but  is  entirely  independent  of  the  shape  of  the  mag- 
netic circuit  and  the  volume  of  iron  used  ;  in  other  words, 
it  is  a  function  of  the  iron  alone  and  not  of  the  electric 
circuit. 

Eddy  currents  have  in  general  no  effect  on  the  shape  of 
the  wave,  but  follow  the  shape  of  the  E.  M.  F.  wave  like  any 
other  secondary  current.  In  their  influence  upon  the  pri- 

*  "  On  the  Law  of  Hysteresis,"  Part  III,  A.  I.  E.  E.  Transactions,  1894, 
p.  595- 


114    HYSTERESIS— MOLECULAR  MAGNETIC  FRICTION. 


mary  current  they  also  act  like  a  secondary  current — that  is, 
in  the  transformer  like  a  partial  load  on  the  secondary  cir- 
cuit, shifting  the  current  wave  ahead  of  the  magnetism,  and 
bringing  it  into  lesser  phase  displacement  from  the  E.  M.  F. 


M 


\ 


\ 


\ 


7 


\ 


M 


\ 


\ 


7 


i 


\ 


£radley  £  Poates,  Engr's,  Jf.f. 

FIGS.  43  AND  44. — DISTORTION  OF  CURRENT  WAVE  BY  HYSTERESIS. 

In  Fig.  43  is  shown  the  current  wave  in  a  transformer  at 
partial  secondary  load,  showing  the  reduction  of  the  distortion 
under  load.  The  dotted  line  M  represents  the  magnetic  fiux. 


HYSTERETIC  LOSSES.  11$ 

In  a  magnetic  circuit  consisting  partly  of  iron  and  partly 
of  air,  the  curve  of  exciting  current  of  an  impressed  sine 
wave  of  E.M.F.  is  the  sum  of  the  exciting  currents  of  the 
iron  and  of  the  air  portions  offthe  circuit.  Since  air  has  no 
hysteresis,  but  very  low  permeability,  the  exciting  current 
of  the  air  portion  of  the  circuit  is  a  sine  wave  of  high  ampli- 
tude, and  therefore,  when  superimposed  upon  the  wave  of 
exciting  current  of  the  iron  circuit,  reduces  the  relative  dis- 
tortion of  the  total  exciting  current,  and  makes  it  appear 
more  sine-shaped  ;  this  is  shown  in  Fig.  44,  which  represents 
the  exciting  current  of  a  magnetic  circuit  containing  an  air- 
gap  of  1/400  of  the  length  of  the  iron,*  and  corresponding 
to  a  maximum  flux  of  (B  =  6000,  or  to  the  curve  shown  in 
Fig.  40.  The  magnetic  flux  is  shown  by  the  dotted  line  M. 

75.  Hysteretic  Losses. — The  loss  of  energy  by  hyster- 
esis depends  upon  the  maximum  value  of  the  magnetic  flux, 
while  the  energy  transformed  in  a  transformer  depends  upon 
the  effective  value  of  E.  M.  F.  With  a  sine  wave  of  E.  M.  F., 
the  magnetic  flux  follows  a  sine  wave  also,  and  therefore  the 
maximum  value  of  the  latter  is  proportional  to  the  impressed 
E.  M.  F.  If,  however,  the  wave  of  impressed  E.  M.  F.  is  dis- 
torted, or  differs  from  the  sine  form,  the  wave  of  magnetic 
flux  will  be  distorted  also  by  the  superposition  of  higher 
harmonics,  and  therefore,  with  the  same  effective  value  of 
impressed  E.  M.  F.,  its  maximum  value  in  the  case  of  a  flat- 
topped  wave  of  magnetic  flux,  can  be  lower  than  with  a  sine 
wave,  or  higher  than  with  a  sine  wave  with  a  peaked  wave, 
of  magnetic  flux.  In  the  first  case  the  same  amount  of 
energy  will  be  transformed  with  a  lesser  hysteretic  loss,  and 
in  the  latter  case  with  a  larger  hysteretic  loss ;  or,  in  other 

*  "On  the  Law  of  Hysteresis,"  Part  III,  A.  I.  E.  E.  Transactions,  1894, 
p.  720. 


Il6     HYSTERESIS—  MOLECULAR   MAGNETIC  FRICTION. 

words,  the  hysteretic  loss  in  a  transformer,  at  the  same  im- 
pressed effective  E.  M.  F.,  depends  upon  the  shape  of  the 
E.  M.  F.  wave,  and  differs  with  different  wave-shapes  of 
E.  M.  F.,  by  as  much  as  30$. 

Iron-clad  alternators  of  "  uni-tooth  "  construction  in  general 
give  peaked  waves  of  E.  M.  F.,  and,  therefore,  flat-topped 
waves  of  magnetic  flux  ;  that  is,  give  hysteretic  losses  in 
transformers  lower  than  with  sine  waves  by  something  like 


Distributed  windings,  that  is,  comparable  to  windings  of 
a  continuous  current  machine  tapped  at  two  diametrically 
opposite  points  of  the  armature,  frequently  give  a  pointed 
wave  of  magnetic  flux,  thus  increasing  hysteretic  losses. 

Since  eddy  currents  are  secondary  currents,  the  loss  of 
energy  by  such  currents  in  transformers  is  entirely  inde- 
pendent of  the  wave-shape  of  impressed  E.  M.  F.,  and  there- 
fore proportional  to  the  load  on  the  secondary  circuit  at 
constant  resistance  ;  or,  in  other  words,  if  the  secondary 
circuit  of  the  transformer  is  closed  by  a  given  resistance,  and 
the  same  effective  E.  M.  F.  impressed  upon  the  primary  cir- 
cuit, the  energy  of  this  E.  M.  F.  will  divide  in  the  same  pro- 
portion between  the  useful  secondary  circuit,  and  the 
secondary  waste  circuit  of  eddy  currents,  independently  of 
the  shape  of  the  E.  M.  F.  wave. 


PROPERTIES  OF  ELECTRIFIED  BODIES.  1 1/ 


ELECTRICITY. 

PROPERTIES    OF    ELECTRIFIED   BODIES. 

76.  Phenomenon  of  Electrification.— When  a  glass  or 
resin  rod  is  rubbed  with  a  piece  of  woollen  cloth,  it  is  ob- 
served that  the  rod  attracts  light  bodies.  The  experiment 
may  be  made  by  suspending  a  ball  of  elder-pith  by  a  silk 
thread.  The  ball  is  first  attracted  by  the  rod,  then  it  is 
repelled  after  having  touched  the  rod.  But  if  a  rod  of 
rubbed  resin  is  brought  near  the  ball,  after  it  has  been 
repelled  by  the  glass  rod,  it  is  observed  that  the  ball  is 
again  attracted. 

The  bodies  between  which  similar  actions  are  shown  are 
said  to  be  electrified,  and  the  name  electricity  is  given  to  the 
unknown  agent  which  produces  these  phenomena. 

Observations  hows  that  the  electric  properties  observed 
in  glass  and  resin  are  general.  Any  two  bodies,  A  and  B, 
attract  each  other  after  having  been  rubbed  together. 
But  two  bodies  of  the  same  nature,  A  and  Ar,  repel  each 
other  after  having  been  rubbed  by  a  third. 

To  interpret  these  phenomena,  opposite  electrifications 
are  attributed  the  bodies  after  rubbing;  those  which  behave 
like  glass,  in  regard  to  wool,  are  said  to  be  positively  electri- 
fied, or  charged  with  positive  electricity;  those  which  act 
like  resin  are  said  to  be  negatively  electrified,  or  charged 
with  negative  electricity.  Electric  actions  are  summed  up 
in  the  following  rule : 

Bodies  charged  with  electricity  of  the  same  name  repel  each 
other  and  attract  those  charged  with  electricity  of  contrary 
name. 


1 1 8  ELE  CTRICIT  Y. 

It  must  be  observed,  however,  that  these  denominations 
do  not  imply  the  existence  of  two  distinct  kinds  of  elec- 
tricity, but  that  they  are  only  forms  of  speech  designed  to 
denote  different  states  of  electrification.  One  must  not 
extend  their  meaning  further  than  is  done  in  mathematics 
in  the  case  of  quantities  of  contrary  sign. 

Following  a  hypothesis  suggested  by  Franklin,  electricity 
is  generally  compared  to  an  imponderable  fluid,  of  which 
every  body  contains  a  normal  quantity.  If  the  charge  sur- 
passes this  quantity,  there  is  positive  electrification.  In  the 
opposite  case  the  electrification  is  negative.  A  body  is  in 
the  neutral  state  when  it  has  its  normal  quantity  of 
electricity.  According  to  some  eminent  physicists,  Clausius 
among  others,  electricity  is  supposed  to  be  the  ether  in 
which  the  molecules  of  all  bodies  are  enveloped  and  which 
fills  interplanetary  space.  The  phenomena  of  electrification 
are,  on  this  hypothesis,  supposed  to  be  due  to  particular 
conditions  of  the  ether. 

77.  Conductors  and  Insulators. — Different  bodies  act 
differently  with  regard  to  electricity.  When  one  of  the 
extremities  of  a  metallic  rod  is  electrified  by  friction, 
attractive  properties  are  immediately  manifested  in  all 
parts  of  the  rod.  In  a  rod  of  resin,  on  the  contrary,  this 
propagation  of  the  phenomenon  is  extremely  slow.  Those 
bodies  which  propagate  electric  actions  rapidly  are  called 
conductors,  the  rest  are  called  insulators. 

Conductors,  which  include  all  metals  and  alloys,  have  to 
be  supported  by  insulators  in  order  to  retain  their  attractive 
properties.  From  the  point  of  view  of  the  experiments 
which  we  are  examining,  the  earth  and  all  bodies  impreg- 
nated or  coated  with  moisture  act  like  conductors.  The 
choice  of  the  insulating  supports  is  therefore  important  for 
the  success  of  these  experiments.  Air  is  frequently  em- 


PROPERTIES  OF  ELECTRIFIED  BODIES.  1 1 9 

ployed  as  an  insulator  ;  in  order  to  prevent  the  moisture  it 
contains  from  condensing  on  the  solid  insulators  which 
support  the  electrified  conductors,  and  thus  producing  a 
moist  conductive  coating,  a  support  may  be  used  whose 
base  passes  through  air  dried  by  concentrated  sulphuric 
acid,  Fig.  45. 


FIG.  45- 
Among    solids  the  insulators  most  commonly  used   are 

glass,  glazed  porcelain,  india-rubber,  ebonite  or  india-rubber 
hardened  by  combination  with  sulphur,  gutta-percha,  paraf- 
fine,  silk,  cellulose,  and  shellac.  Some  of  these  substances, 
such  as  glass  and  cellulose,  are  hygroscopic.  They  should 
be  covered  with  a  coating  of  insulating  varnish  or  im 
pregnated  with  paraffine. 

78.  Electrification  by  Influence. — The  experiment  with 
the  pith-ball,  §  76,  has  shown  us  that  a  body  can  be  electri- 
fied by  contact.  Even  the  mere  approach  of  an  electrified 
body  is  sufficient  to  produce  electrical  manifestations  in  a 
neighboring  conductor. 

Thus  a  sphere,  A,  charged  with  positive  electricity,  being 
brought  over  a  conductor  BC,  Fig.  46,  it  is  observed  that 
the  extremities  of  this  latter  act  upon  a  freely  suspended 
pith-ball.  If  this  ball  be  charged  beforehand  with  positive 
electricity,  it  is  seen  to  be  attracted  by  the  extremity  B,  and 
repelled  by  C.  From  this  we  conclude  that  the  former  is 
negatively,  the  latter  positively,  electrified.  When  the 
influencing  sphere,  A,  is  withdrawn,  the  conductor,  BC,  no 
longer  acts  on  a  ball  in  the  neutral  state  ;  which  shows  that 


120 


ELECTRICITY. 


the   contrary   electricities    accumulated    in   B  and   C  have 
re-established  the  neutral  state  by  recombining. 


FIG.  46. 

If  the  conductor  BC  is  connected  for  a  moment  to  the 
earth  while  it  is  under  the  influence  of  A,  we  observe  after 
taking  away  A  that  every  portion  of  BC  is  charged 
negatively. 

This  method  of  charging  the  conductor  is  called  electri- 
fication by  influence. 

79.  Let  us  suppose  that  the  framework  of  a  quadrant 
electrometer,  E,  Fig.  47,  be  connected  to  the  earth  through 


FIG.  47- 

the  suspension-wire,  and  that  the  two  pair  of  quadrants  are 
electrified,  but  of  opposite  sign.  The  framework  will  be 
electrified  by  influence ;  it  will  be  equally  attracted  in  both 


PROPERTIES  OF  ELECTRIFIED  BODIES. 


121 


directions  and  will  remain  motionless.  But  if  it  receives  a 
positive  charge,  it  will  immediately  be  displaced  towards 
the  negative  quadrants.  If  the  charge  be  negative,  the 
displacement  will  be  towards  '"the  positive  quadrants.  We 
shall  show,  further  on,  that  tfete  swing  of  the  framework  is 
proportional  to  the  charge  that  has  been  given  to  it ;  we 
shall  also  see  that  in  order  to  give  the  quadrants  equal  and 
contrary  charges  they  need  only  be  connected  to  the  poles 
of  a  battery. 

This  apparatus,  which  is  suitable  for  measuring  feeble 
electrifications,  will  enable  us  to  proceed  to  some  funda- 
mental experiments. 

80,  Experiments. — Let  us  consider  a  metallic  cylinder, 
A,  Fig.  48,  open  at  the  top  and  supported  on  an  insulating 


FIG.  48. 

base ;  let  us  also  connect  the  cylinder  with  the  framework, 
c,  in  the  electrometer  by  a  thin  wire,  both  pairs  of  quadrants 


122  ELECTRICITY. 

being  electrified  in  the  manner  indicated  in  the  preceding 
paragraph. 

If    the  cylinder  be  in  the    neutral  state,  the  framework 
will  not  change  its  position. 

I.  Introduce   into  the  cylinder  a  conductor,  B,  hung  from 
a  silk  thread  and  charged  with  positive  electricity.    We  know, 
by  the  phenomenon  of  influence,  that  the  internal  wall  of  the 
cylinder  will  be  charged  negatively,  and  that  the  electricity 
of  the  same  sign  as  that  of  B  will  be  repelled  towards  the  ex- 
ternal wall  and  to  the  framework,  c.    This  latter  will  exhibit 
a  displacement  which  will  increase  as  the  conductor  B  de- 
scends,  until  this  last  has  reached  a  certain  depth  in  the 
cylinder.     From  this  point  onwards  the  displacement  of  the 
framework  c  remains  invariable,   whatever  be  the  position 
of  B;  even  if  B  be  put  in  contact  with  the  cylinder,  the  dis- 
placement of  the  electrometer  remains  the  same.      Upon 
withdrawing  the   body   after  contact  it  will  be  found,   by 
means  of  a  pith-ball  electroscope  or  a  second  quadrant  elec- 
trometer, that  B  has  returned  to  the  neutral  state.     Hence 
we  conclude  that   the   positive  charge  which   it   has  parted 
with  to  the  cylinder  has  been   neutralized   by  the  negative 
charge  developed  by  influence,  the  cylinder  preserving  its 
induced  positive  charge. 

If    the   charge   on    B    were    negative,    the    electrometer 
would  have  shown  a  displacement  in  the  opposite  direction. 

II.  Place    inside  the   cylinder  a  metallic  sphere  charged 
with  electricity  and  giving  a  displacement  a.     Touch  this 
sphere  with  an  equal  one  in  the  neutral  state.  Then  the  two 
spheres,  if  placed  successively  inside  the  cylinder,  will  give 
deviations  equal  to  a/2. 

III.  Suspend   inside  the  cylinder  two  insulated  bodies  in 
the  neutral  state  and  rub  them  one  against  the  other.     The 
electrometer  will  show  no  deviation,  but  will  remain  in  its 
first  position. 


PROPERTIES  OF  ELECTRIFIED  BODIES.  12$ 

IV.  If  we  introduce  separately  into  the  cylinder  various 
bodies,  B,  D,  E,  charged  with  electricity,  we  get  deviations 
some  of  which  are  positive,  others  negative,  according  to  the 
signs  of  the  charges.  ,  " 

On  introducing  simultaneously  B,  D,  and  E  the  devia- 
tion obtained  is  the  algebraic  sum  of  the  preceding  devia- 
tions and  remains  invariable,  even  if  the  bodies  are  put  in 
contact  or  rubbed  against  each  other. 

From  these  experiments  we  conclude  that  electric 
charges  are  quantities  susceptible  of  measurement.  We 
might  take  a  given  charge  as  unity  and  consider  as  double 
and  triple  those  charges  which  produce  a  double  or  triple 
deviation  in  the  electrometer  connected  with  the  cylinder. 

The  two  last  experiments  show  that  the  total  charge  of 
a  system  of  electrified  bodies  is  invariable,  and  that  friction 
produces  on  bodies  equal  and  opposite  electricities,  capable 
of  neutralizing  each  other. 

If,  for  example,  we  electrify  a  resin  rod  by  aid  of  a  piece 
of  cloth,  a  quantity  of  electricity  equal  and  opposite  to  the 
charge  on  the  resin  is  produced  on  the  cloth  and  on  the 
neighboring  conductors  connected  with  it,  such  as  the  body 
of  the  experimenter,  the  table  used  for  the  experiment,  and 
the  walls  of  the  room. 

81.  In  the  case  of  a  conductor  in  equilibrium  the  elec- 
tricity is  distributed  on  its  surface.  —  This  important 
property  can  be  demonstrated  in  various  ways. 

I.  For  example,  let  A,  Fig.  49,  be  a  hollow  metallic 
sphere.  Touch  the  exterior  surface  of  the  sphere  with  a 
proof-plane  formed  of  a  disc  of  copper  foil  on  an  insulating 
handle ;  the  disc  will  take  an  electric  charge,  as  can  be  dis- 
covered by  means  of  an  electroscope  or  electrometer.  If, 
on  the  other  hand,  the  point  touched  be  on  the  interior  sur- 
face of  the  sphere,  the  disc  will  show  no  electrification. 


124  ELECTRICITY. 

The  method  of  the  proof-plane,  though  by  no  means  ex- 
act, can  be  employed  to  compare  the  electric  charges  on  the 
different  parts  of  a  body.  If  we  touch  successively  various 
points  on  the  surface  of  an  electrified  sphere,  we  find  that 


FIG.  49. 

the  charge  carried  away  each  time  is  constant,  that  is,  that 
the  sphere  is  uniformly  charged.  For  a  body  of  ovoid  form 
we  find  that  the  charge  increases  inversely  as  the  radius  of 
curvature  of  the  surface. 

II.  The  fact  of  the  external  distribution  of  the  charge  on 
a  conductor  has  been  established  beyond  doubt  by  Faraday, 
who  had  a  chamber  four  metres  square  constructed,  sup- 
ported on  insulators  and  covered  with  metallic  sheets.  He 
entered  this  chamber  while  it  was  electrified  and  was  unable 
to  discover  the  least  trace  of  electricity  on  its  internal  walls, 
though  using  the  most  delicate  electrometers. 

82.  Law  of  Electric  Actions. — The  preceding  facts,  on 
being  summed  up,  show  that  the  action  between  electrified 
bodies  must  be  placed  amongst  the  central  forces,  and  that 
we  can  define  by  quantity  of  electricity,  or  electric  mass,  or 
charge  of  a  body,  a  quantity  proportional  to  the  force  which 
it  exercises  upon  neighboring  electrified  bodies. 

The  superficial  density  is  the  charge  per  unit  surface. 

In  virtue  of  this  definition  the  elementary  law  of  the 
force  between  two  electric  masses  q  and  qf ,  at  a  distance  /, 
will  be  of  the  form 

/=  kqq'&l}. 


PROPERTIES  OF  ELECTRIFIED  BODIES.  125 

The  function  0(/)  is  determined  by  the  fact  that  the 
electric  action  is  zero  in  the  interior  of  an  electrified  con- 
ductor in  equilibrium,  §81. 

The  reciprocal  of  the  theorem  demonstrated  in  §  26 
shows  that,  in  order  th#t  a  homogeneous  spherical  shell  may 
have  no  action  upon  an  internal  point,  the  force  must  be  in 
inverse  ratio  to  the  square  of  the  distance. 

The  law  of  electric  attraction  and  repulsion  is  therefore 


in  which  we  consider  the  force  f  as  repulsive  or  attractive 
according  as  the  masses  q,  q'  are  of  the  same  or  contrary 
sign. 

This  law,  experimentally  demonstrated  by  Coulomb,  en- 
ables us  to  apply  to  electric  attraction  the  general  properties 
of  the  Newtonian  central  forces. 

83.  Definitions.    Electric  Field.     Electric  Potential.— 

By  electric  field  we  will  designate  a  space  or  region  where 
electric  forces  take  origin.  The  intensity  in  a  point  of  an 
electric  field  is  the  resultant  of  the  forces  exercised  there 
upon  a  positive  mass  taken  as  unity.  The  direction  of  this 
resultant  is  called  the  direction  of  the  field. 

Electric  forces  are  defined  by  aid  of  a  potential,  §  12, 
called  electric  potential,  which  has  as  its  expression 


The  intensity  of  the  field  in  a  direction  s  is  expressed  as 
a  function  of  the  potential  by 

K      da 

X'  =    "57- 

84.  Potential  of  a  Conductor  in  Equilibrium.—  Since  the 

force  is  zero  at  the  interior  of  a  conductor  carrying  an  elec- 


126  ELECTRICITY. 

trie  charge  in  equilibrium,  §  81,  the  electric  potential  has 
the  same  value  at  all  internal  points  ;  the  surface  of  the  con- 
ductor is  equipotential,  and  the  lines  of  force  are  normal  to 
its  surface. 

This  constant  value  is  called  the  potential  of  the  con- 
ductor. 

The  idea  of  potential  may  be  distinguished  from  the 
idea  of  charge  by  the  following  experiment  ;  The  method 
of  the  proof-plane  demonstrates  that  the  charge  is  variable 
at  different  points  of  a  conductor  of  irregular  shape.  Nev- 
ertheless, if  we  successively  connect  these  points  by  a  wire 
with  the  electrometer,  the  deviation  remains  constant,  for 
the  force  which  urges  the  electricity  to  pass  from  the  con- 
ductor to  the  electrometer  depends  only  on  the  potential, 
which  is  invariable. 

In  the  case  of  a  metallic  sphere  of  radius  R  electrified  to 
a  density  cr  the  potential  at  the  centre  is 


26. 


K  K. 


This  quantity  is  the  potential  of  the  sphere. 

85.  Potential  of  the  Earth.  —  Electric  forces  do  not 
depend  at  all  on  the  absolute  values  of  the  potential,  but 
only  on  its  variation.  Thus  the  force  which  impels  the 
electricity  of  a  conductor  to  flow  to  an  adjoining  conductor, 
such  as  the  earth,  depends  on  the  difference  between  the 
potentials  of  the  bodies  under  consideration  ;  positive  elec- 
tricity tends  to  be  displaced  in  the  direction  of  decreasing 
potentials.  The  ground  and  the  walls  of  an  experimenting- 
room  being  conductors,  there  is  no  objection  to  considering 
their  potential  as  zero  and  then  taking  as  positive  those 
potentials  above  that  of  the  earth,  the  potentials  being  nega- 
tive in  the  opposite  case. 


PROPERTIES  OF  ELECTRIFIED  BODIES.  I2/ 

86.  Coulomb's  Theorem.  —  The  intensity  of  the  field  at  a 
point  infinitely  near  a  conductor  in  equilibrium  is  equal  to  ^nk 
multiplied  by  the  superficial  density  at  that  point. 

Let  us  consider  an  element  ds  of  the  surface  charged 
with  a  quantity  of  electricity  e^ual  to  ads,  a  being  the  den- 
sity. Let  us  suppose  a  tube  of  force,  passing  through  the 
contour-line  of  ds,  limited  externally  by  an  infinitely  near 
equipotential  surface  U,  and  closed  inside  the  conductor  by 
any  surface  5.  Let  us  then  apply  Gauss's  theorem,  §  20,  to 
the  volume  thus  enclosed.  The  flux  of  force  is  zero  across 
the  surface  5,  since  there  is  no  electric  force  inside  a  con- 
ductor in  equilibrium  ;  as  there  is  no  component  across  the 
lateral  walls  of  the  tube,  the  outflowing  flux  from  the  given 
volume  is  limited  to  the  flux  across  the  equipotential  sur- 
face Ur 

Let  OCg  be  the  intensity  of  the  field  normal  to  this  surface  ; 
the  flux  3£eds  is  equal  to  ^nk  multiplied  by  the  quantity  of 
electricity  ads. 

Consequently 

OC,  ==  47tk<r. 

The  intensity  of  the  field  having  the  same  sign  as  the 
density,  it  is  directed  outwards  for  a  positive  density,  inwards 
fcr  a  negative  destiny. 

We  also  have 


U  being  the  potential  of  the  conductor,  n  a  direction  normal 
to  the  surface. 

We  see  that  the  property  demonstrated  in  the  case  of  a 
sphere,  §  27,  is  general  for  all  electrified  conductors  in 
equilibrium. 


ELECTRICIT  Y. 


87.  Electrostatic  Pressure.—  The  method  of  reasoning 
applied  to  the  end  of  §  30  shows  that  the  force  exercised  by 
the  whole  charge  of  a  conductor  in  equilibrium  upon  the 
electricity  covering  unit  surface  is  equal  to  2nk  multiplied 
by  the  square  of  the  density  at  the  given  point  : 


P= 

This  force,  called  electrostatic  pressure,  is  always  positive 
whatever  be  the  sign  of  a.  Hence  it  results  that  the  elec- 
tricity has  a  tendency  to  escape  from  the  conductor  and 
enter  the  surrounding  medium.  If  the  conductor  is  mova- 
ble, it  may  be  moved  in  the  direction  of  the  maximum 
pressure.  Thus,  when  two  conductors  are  electrified  with 
opposite  signs,  their  electric  density  is,  by  the  phenomenon 
of  induction,  greater  on  the  neighboring  faces  than  on  the 
opposite  ones  ;  the  electrostatic  pressure  therefore  tends  to 
urge  the  conductors  towards  each  other. 

When  the  bodies  are  charged  with  electricity  of  the  same 
sign,  the  maximum  densities  are  on  the  outer  faces  and  the 
resultant  pressures  are  in  such  a  direction  as  to  move  the 
conductors  apart.  So,  too,  if  a  soap-bubble  be  electrified,  it 
expands  until  the  increased  surface-tension  of  the  liquid 
layer  balances  the  electrostatic  pressure. 

88.  Corresponding  Elements.  —  Suppose  a  tube  of 
force,  traversing  an  electric  field,  be  limited  by  two  electri- 
fied conductors  on  which  it  cuts  surfaces  s,  s'  called  corre- 
sponding elements.  Let  q  and  q'  be  the  charges  of  these 
surfaces.  Then,  applying  Gauss's  theorem,  §  20,  to  the  vol- 
ume limited  by  the  tube  and  by  any  two  surfaces  inside  the 
conductors,  it  is  easy  to  see  that  the  outflowing  flux  is  zero, 
since  the  electric  force  has  no  component  inside  the  con- 
ductors n.or  across  the  lateral  walls  of  the  tube. 


Pit  OPER  TIES  OF  ELE  CTR1FIED  B  OD  IES. 

Consequently 


whence 


, 
=  -q. 


We  conclude  from  this  that  corresponding  elements  carry 
equal  and  opposite  quantities  of  electricity.  The  lines  of  force 
uniting  them  take  their  origin  from  the  points  charged  with 
positive  electricity  and  end  in  points  covered  with  negative 
electricity. 

The  lines  of  force  which  leave  an  electrified  body  neces- 
sarily find,  therefore,  an  opposite  electric  charge  in  their 
vicinage.  This  charge  is  spread  over  the  neighboring  con- 
ductors, over  the  table  which  supports  the  body,  over  the 
walls  of  the  experimenting-room.  This  is  the  true  method 
of  looking  at  the  phenomenon  of  induction,  §  78,  the  lines 
of  force  which  emanate  from  the  inducing  body  impinging 
on  the  influenced  body. 

89.  Power  of  Points.  —  If  the  inducing  body  is  furnished 
with  a  point,  the  lines  of  force  emerging  from  this  latter 
form  a  conical  tube.     The  apex  of  the  cone  carries  a  charge 
equal  to  that  of  the  base,  so  that  the  density  acquires  an 
excessive  value  at  the  point,  and  the  electrostatic  pressure 
urges   the   contrary   electricities   to   recombine  across   the 
medium  which  separates  them. 

90.  Electric  Screen.  —  A    hollow  conductor  surrounded 
by  electrified  bodies  will  assume  on  its  surface  a  distribution 
of  induced  electricity;  its  potential  will  acquire  a  constant 
value  equal  to  the  potential  of  all  the  internal  points,  §  84. 
Consequently  no  electric  force  will  exist  inside  the  conduct- 
or, and  any  objects  that  it  may  contain  will  be  entirely  cut 
off  from  the  action  of  the  exterior  charges.  The  hollow  con- 


1 30  ELECTRICIT  Y. 

ductor  thus  serves  as  a  screen  to  the  bodies  it  envelops. 
Faraday's  experiment,  §  81,  is  a  confirmation  of  this 
property. 

Maxwell  has  shown  that  the  screen  need  not  necessarily 
be  continuous ;  a  metallic  gauze  constitutes  an  efficacious 
screen. 

Similar  means  of  protection  are  used  to  cut  off  electrom- 
eters from  external  influences. 

91.  Lightning-rods. — The  two  properties  just  considered 
are  made  use  of  in  the  preservation  of  buildings  from  light- 
ning.    The  best  lightning-rod  would  be  a  metallic  covering 
completely  enveloping  the  whole  building.     As  this  is  hardly 
practicable,   we  may  adopt,  as  was    done    by   Melsens  (in 
Brussels),  the   plan  of  covering  the  building  with  a  network 
of  conductors  connected  with  all  the  metallic   portions   of 
the  building. 

This  network  is  connected  to  earth  by  means  of  the  gas- 
and  water-pipes  and  by  metal  plates  sunk  in  ponds  or  moist 
earth.  On  the  top  points  of  the  building  are  fixed  pointed 
rods  or  clusters  of  spikes  of  metal  connected  to  the  general 
network. 

CONDENSERS— DIELECTRICS. 

92.  Capacity  of  Conductors. — Suppose  an  insulated  con- 
ductor A  to  be  in  the  neighborhood  of  other  conductors  B, 
C,  D,  maintained  at  a  constant  potential  by  some  arrange- 
ment, for  example,  by  connection  with  the  earth.    If  we  give 
A  an  electric  charge  Q,  the  other  conductors  assume  induced 
charges  which  are  distributed   according  to  a  law  depending 
on  the  form  of  these  bodies,  on  their  relative  positions,  and, 
as  we  shall  see,  on  the  medium  which  separates  them.     The 
sum    of   these   charges  is   equal  and  contrary  to  Q.     The 
potential  of  A  will  be  expressed  by 


PROPERTIES  OP  ELECTRIFIED  BODIES.  131 

If  we  increase  the  charge  on  A,  the  induced  charges  in- 
crease in  the  same  proportion  ;  the  new  induced  layers  are 
superposed  on  the  first,  and  if  the  relative  distances  remain 
the  same  the  potential  of  *A  grows  in  proportion  to  its 
charge. 

Consequently  the  ratio  between  the  charge  of  A  and  its 
potential  is  a  constant  which  only  depends  on  the  shape  and 
relative  position  of  the  conductors  composing  the  system, 
and  on  the  medium  which  separates  them. 

This  constant, 

C-Q 

-  u' 

is  called  the  capacity  of  the  conductor  A. 

If  U  =  i,  C  =  Q\  the  capacity  of  a  body  is  therefore 
measured  by  the  charge  which  raises  its  potential  one  unit 
above  the  potential  of  the  surrounding  conductors.  Never- 
theless the  capacity  is  not  a  charge,  any  more  than  the  con- 
tent of  a  vessel  for  holding  a  liquid  represents  a  quantity  of 
the  liquid. 

93.  Condensers. — Spherical  Condenser. — As  an  appli- 
cation of  the  preceding,  let  us  consider  a  metallic  sphere 
enveloped  by  a  concentric  hollow  sphere,  this  latter  being 
connected  to  earth. 

Such  an  arrangement  constitutes  a  condenser,  the  spheres 
being  the  armatures  (or  plates)  and  the  insulator  which 
separates  them  the  dielectric. 

When  a  charge  -\- q  is  given  to  the  interior  sphere,  the 
internal  wall  of  the  other  sphere  takes,  by  induction,  a 
charge  equal  and  opposite  to  the  first,  §  88,  and  the  electric- 
ity of  the  same  name  as  that  of  the  interior  sphere  escapes 
to  the  earth. 

If  we  denote  by  /  and  I'  the  radii  of  the  two  electric  layers, 


132  ELECTRICI T  Y. 

the  potential  at  the  centre  of  the  two  spheres  is 

According  to  the  definition  given  above,  §  92,  the  capacity 
of  the  interior  sphere,  called  also  the  capacity  of  the 
condenser ',  is 

C-  q--     — - 
"  U~  k(l'  —I)' 

We  see  that  the  capacity  of  a  condenser  increases  in- 
versely to  the  distance  between  its  plates. 

If  the  radius  I'  were  to  increase  indefinitely,  we  should  have 
at  the  limit  an  insulated  sphere  whose  capacity  would  be 

C=l' 

94.  Plate  Condenser. — Let  us  take  the  case  of  a  con- 
denser formed  of  two  parallel  discs,  with  rounded  edges,  A, 
B,  A  being  electrified,  while  B  is  connected  with  the  earth. 
The  adjoining  faces  of  A  and  B  take  equal  and  opposite 
charges  of  electricity,  and  the  lines  of  force  run  perpendicu- 
larly from  one  face  to  the  other,  except  at  the  edges,  where 
the  lines  curve  in  such  a  way  as  to  remain  perpendicular  to 
the  surfaces.  We  can  therefore  admit  that  in  the  central 
region  the  electric  field  is  uniform. 

The  intensity  of  the  field  is  expressed  by 


n  being  the  direction  of  a  line  of  force.  As  the  intensity  is 
constant  in  this  direction,  we  have,  representing  by  U  the 
potential  of  A  and  by  /  the  distance  between  the  plates, 


OC,   /  dn  =     I  — 

c/O  J U 


dU; 


PROPERTIES  OF  ELECTRIFIED  BODIES.  133 

whence 

W,el=U  and   OC,  =  y. 

But  the  field  is  also  expressedby  JC,  =  ^nk<j,  §  86,  a  being 
the  surface-density. 
Consequently 

U 


whence 

U 


(7  = 


The  charge  on  a  surface  s,  taken  in  the  middle  region, 
will  be 

Us 


The  capacity  of  the  condenser  referred  to  this  surface  will 
be  expressed  by 


We  have  neglected  the  charge  taken  by  the  plate  A  on  its 
posterior  face  ;  but  if  /  is  small  this  quantity  is  negligible 
compared  with  that  on  the  anterior  surface. 

A  condenser  of  great  capacity  is  obtained  by  superposing 
leaves  of  tinfoil,  separated  by  sheets  of  paraffined  paper  or 
of  mica  and  connected  in  two  series,  the  first  comprising 
the  even-number  sheets,  the  second  comprising  the  odd- 
number  ones.  The  sheets  of  such  a  condenser  cannot  be 
charged  at  very  different  potentials  for  fear  of  piercing  the 
dielectric  by  sparking. 

95.  Guard-ring  Condenser.  —  Lord  Kelvin  has  devised 
a  uniformly  charged  plane  plate  by  cutting  a  narrow  circu- 
lar groove  through  the  plate,  A,  Fig.  50,  and  electrically 


134 


ELECTRICITY. 


connecting   the  interior  disc  with  the  exterior  ring,  called 
guard-ring,  by  a  slender  wire.     It  can  be  admitted  that  the 


FIG.  50. 

circular  cut  has  not  sensibly  modified  the  uniformity  of  the 
field,  so  that  the  capacity  of  the  disc  A,  of  surface  s,  is 
rigorously 


96.  Absolute  Electrometer.  —  The  pressure  which  is  ex- 
ercised  by  unit  surface  on  the  disc  A  is,  §  87, 


the  pressure  on  a  surface  s  will  be 

p  =  27tko*s. 


U 


<T  = 


Now 


whence 


If  we  support  the  disc  A  by  the  beam  of  a  balance,  Fig. 
50,  we  will  be  able  to  balance  the  attraction  of  the  two  discs 
by  a  known  weight.  Expressing  this  weight  in  dynes,  we 


ITs 


PROPERTIES  OF  ELECTRIFIED  BODIES.  135 

can  deduce  from  the  preceding  equation  the  difference  of 
potential  of  the  two  plates : 


The  above  is  the  principle  of  Lord  Kelvin's  absolute 
electrometer. 

97.  Cylindrical  Condenser.  —  Suppose  a  condenser  formed 
of  two  indefinite  concentric  cylinders.  The  interior  cylinder, 
of  radius  rlf  having  a  charge  q,  the  outer  cylinder,  of  radius 
rs,  connected  to  the  earth,  will  take  a  charge  —  q.  Such  a 
condenser  is  realized  in  practice  by  a  conducting  wire 
covered  with  an  uniform  insulating  layer  and  submerged  in 
water,  this  latter  serving  as  the  outer  plate. 

The  dielectric  can  be  divided  by  equipotential  surfaces 
which  are  concentric  with  the  conductor  by  reason  of 
symmetry.  The  tubes  of  force  are  limited  by  planes  passing 
through  the  axis  of  the  cylinders. 

Let  us  take  the  case  of  a  tube  of  opening  a,  limited  by 
two  planes  normal  to  the  axis  and  one  centimetre  apart. 

The  flux  of  force  is  constant  in  the  tube,  §  21. 

Denoting  by  s  a  section  of  the  tube  by  an  equipotential 
surface  of  radius  r,  we  have 

OC.J  —  OC,  X  ar  =  const.  , 

whence,  denoting  by  5CX  the  intensity  of  the  field  infinitely 
near  the  interior  plate,  we  get 


l  ,  §  86. 
The  capacity  per  unit  of  length  is 


1 3°  ELECTRICIT  Y. 

But 


whence 


r~AU=    /\dr  =    r 

Ju  Jri  Jn 


which  gives 

U  =  ^7tkcrrl  log,  — *  ; 
*\ 

and  consequently 


U 


This  expression  shows  that,  when  we  connect  condensers 
with  a  source  of  electricity,  we  must  avoid  using  insulated 
wires  in  contact  with  conducting  surfaces,  for  a  conden- 
sation is  then  produced  between  the  wires  and  the  surfaces 
which  may  make  our  results  incorrect.  It  is  best  to  employ 
slender  wires,  at  a  considerable  distance  from  conducting 
surfaces,  so  as  to  obtain  a  negligible  capacity. 

98.  In  the  preceding  calculations  we  have  supposed  one 
of  the  condenser-plates  to  be  connected  to  earth,  so  that  it 
preserves  a  constant  potential,  which  in  this  case  is  zero. 
If  the  plates  were  given  potentials  £/,  and  £/, ,  the  capacity 
would  be  expressed  by 

r         q 
~~~fj — 77" 

99.  Leyden  Jar. — If  in  the  formula  for  a  spherical  con- 
denser we  suppose  the  radii  r  and  r'  but  slightly  different, 
we  have  approximately 

r* 


PROPERTIES  OF  ELECTRIFIED  BODIES.  1  37 

b  being  the  distance  between  the  spheres  ;  whence 


p 

This  formula,  identical  with  that  for  a  plane  condenser,  is 
applicable  to  a  condenser  of  any  form  whatever,  provided 
that  the  plates  be  sufficiently  near  together  and  their 
curvature  sufficiently  slight  for  us  to  consider  the  field 
inside  the  dielectric  as  uniform. 

When  great  exactness  is  unnecessary,  this  is  allowed  to  be 
the  case  in  the  Leyden  jar.  Contrary  to  the  tinfoil  con- 
denser, described  in  §  94,  this  condenser  can  sustain  con- 
siderable differences  of  potential.  It  consists  of  a  bottle  or 
jar  of  glass  varnished  with  shellac,  and  whose  inner  and 
outer  surfaces  are  covered  with  tinfoil  up  to  a  certain 
distance  from  the  mouth  of  the  jar.  A  metal  rod,  passing 
through  the  neck  and  terminating  in  a  knob,  gives  a  means 
of  connecting  the  inner  coating  with  a  source  of  electricity. 

In  order  to  obtain  great  capacity  a  number  of  jars  can  be 
connected  together,  the  inside  coatings  being  joined  to 
each  other,  and  the  outside  coatings  also  connected  by 
themselves. 

ioo.  Energy  of  Electrified  Conductors.  —  Let  q,  q',  q".  .  . 

be  a  system  of  electric  masses  at  potentials  U,  U'  ,  U"  .  .  . 

In  virtue  of  the  properties  demonstrated  for  Newtonian 
forces,  §  25,  the  potential  energy  of  the  system  is 

W=  V2qU. 

It  will  be  noticed  that  the  conductors  connected  to 
earth,  whose  potential  is  zero,  do  not  furnish  any  element 
to  the  above  sum,  even  when  they  carry  electric  charges. 

In   the  case  of  a   condenser  one   plate  of  which  is  con- 


I  38  ELECTRICIT  Y. 

nected  to  earth,  q  and  U  being  respectively  the  charge  and 
potential  of  the  other  plate, 


If  we  denote  by  C  the  capacity  of  the  condenser, 

q=  CU; 
whence 


101.  Theory    of   the    Quadrant    Electrometer.—  The 

cylindrical  framework  of  the  quadrant  electrometer  (which 
for  the  sake  of  simplicity  we  shall  call  the  needle  of  the 
electrometer),  Fig.  51,  constitutes  a  condenser  with  each  of 
the  pairs  of  quadrants,  these  latter  being  connected  two 
and  two. 


FIG.  51. 

Call  U  the  potential  of  the  needle, 
£7,  that  of  the  quadrants  A,  B, 
£7,  that  of  the  quadrants  C,  D. 

The  charge  of  the  condenser  E  —  A,  B  is  proportional 
to  the  difference  of  the  potentials  U—  £7, ,  and  the  horizontal 
component  of  the  force  tending  to  displace  the  needle 
towards  the  given  quadrants  can  be  represented  by 
a(U  —  £7,)2,  a  being  a  constant  coefficient,  §  87. 

The  needle  is  drawn  in  the  opposite  direction  by  a  force 
equal  to  a(U  —  £7a)a,  the  constant  a  being  the  same,  in  con- 


PROPER  TIES  OF  ELECTRIFIED  BODIES.  1  39 

sequence  of  the  slight  displacement  of  the  needle  and  the 
identity  of  shape  of  the  quadrants. 
The  resultant  of  these  two  actions  is 


a(U,-  U,)(W-  U,-  U,). 

& 

Its  moment  is  counterbalanced  by  the  torsion-couple  of 
the  suspending  wire.  Coulomb  has  shown  that,  within 
certain  limits,  this  couple  is  proportional  to  the  angle  of 
torsion. 

We  may  therefore  write 


This   formula  shows  that  the  deviation  is  zero  when  the 
quadrants  are  at  the  same  potential. 

102.  I.  When  the  potential  of  the  needle   is  very  large 
relatively  to  that  of  the  quadrants,  the  formula  reduces  to 

ft  =  6(U,  -  U,)U. 

II.  If   we   connect    the    needle   to   one  of   the   pairs   of 
quadrants,  CD,  for  example,  we  have 

U=U,; 
whence 


The  deviation  is  then  proportional  to  the  square  of  the 
difference  of  potential  of  the  needle  and  A,  B. 

III.  In  the  case  where  the  quadrants  are  maintained  at 
equal  and  opposite  potentials,  6^  —  —£/,,  which  can  be  done, 
as  we  shall  see,  by  connecting  them  to  the  poles  of  a  battery, 
we  have 

ft  =  2bUU>. 

The  deviation  is  proportional  to  the  potential  of  the 
needle. 


140  ELECTRICITY. 

M.  Gouy*  has  shown  that  the  preceding  formula  does 
not  hold  when  the  potentials  of  the  quadrants  are  consider- 
able relatively  to  that  of  the  needle.  In  such  case  there 
exists  a  directive  electric  couple  acting  together  with  the 
torsion-couple,  and  whose  action  must  be  taken  into  account. 
It  must  be  further  remarked  that,  if  the  needle  is  not  of 
the  same  metal  as  the  quadrants,  there  arises  a  difference  of 
potential  by  contact,  §  107,  which  slightly  alters  the  results, 
and  which  is  eliminated  by  making  two  successive  experi- 
ments in  which  this  extraneous  difference  of  potential  pro- 
duces equal  and  opposite  effects.  The  mean  of  the  devia- 
tions thus  obtained  is  the  result  desired. 

103.    Specific    Inductive    Capacity    of    Dielectrics.  — 

We  have  so  far  paid  no  attention  to  the  dielectric  separating 
the  condenser-plates.  For  the  purpose  of  studying  this  ele- 
ment let  us  consider  two  identical  plane  condensers,  the 
dielectric  in  the  one  being  air,  and  in  the  other  paraffine. 

The  similar  plates  being  connected  to  earth,  electrify  the 
free  plate  of  the  first  condenser  to  the  potential  U,  and  con- 
nect it  with  the  needle  of  an  electrometer  whose  quadrants  are 
supposed  to  be  at  equal  and  contrary  potentials.  Let  a  be 
the  deviation  of  the  needle.  Then  connect  the  free  plates  of 
the  two  condensers  by  a  wire.  As  the  initial  charge  q  spreads 
over  both  condensers  the  potential  diminishes  and  becomes 
£/';  the  deviation  of  the  electrometer  falls  to  a'  .  Calling  c 
and  c'  the  capacities  of  the  two  condensers,  we  get  the 
condition 


whence 

U 

U' 


Journal  de  Physique,  1888. 


PROPERTIES  OF  ELECTRIFIED  BODIES.  14! 

But  jj-,  =  -^;  consequently 


The  ratio  —  is  called  the  Specific  inductive  capacity  of  the 

paraffine  in  comparison  with  air.     This  ratio  is  about  2.3  in 
the  present  case. 

We  might  compare  the  capacities  of  dielectrics  to  that 
of  a  vacuum  by  placing  one  of  the  condensers  in  a  vacuum- 
chamber.     We  give  here  some  of  the  values  thus  obtained, 
according  to  Boltzman: 

Sulphur  ........................    3.84 

Glass  ...........................    5.83  to  6.34 

Paraffine  ......................   2.32 

Ebonite  ........................   2.2  1  to  2.76 

Essence  of  turpentine  ...........   2.21 

Air  ..........  .  ...................  1.00059 

Carbonic  anhydride  .............    1.000946 

The  comparison  of  these  values  with  the  indices  of  re- 
fraction i  and  i'  of  the  same  substances  for  light  shows  that 
the  former  are  sensibly  proportional  to  the  squares  of  the 
latter. 

As  the  index  of  refraction  of  a  medium  is  inversely  pro- 
portional to  the  velocity  of  light  in  it,  it  follows  that  the 
specific  inductive  capacities  of  bodies  are  inversely  propor- 
tional to  the  square  of  the  velocity  of  light  in  those  bodies  : 


104.  Nature  of  the  Coefficient  k  in  Coulomb's  Law.  — 

Given  an  air-condenser  charged  with  a  quantity  of  electricity 
q  at  a  potential 

V=k?l 


142  ELE  CTRICI T  Y. 

the  capacity  is 

•=*• 

Let  us  substitute  for  air  some  other  dielectric  ;  keeping  the 
charge  constant,  we  will  get  a  new  capacity, 


the  potential  necessarily  varying  and  becoming 

;  ;:^r     ^4 

Hence  we  deduce 

L  -  —  -  -  -  — 
7~~  u  ~  k  ~ '  ~tf° 

Now  the  ratio  of  the  capacities  —  is  the  same  as  that  of 

the  specific  inductive  capacities. 

Consequently  the  coefficient  in  Coulomb's  law  is  inversely 
proportional  to  the  inductive  capacity  of  the  dielectric  sep- 
arating the  electrified  bodies,  and,  if  the  deductions  of  the 
last  paragraph  are  true,  this  coefficient  is  also  proportional 
to  the  square  of  the  velocity  of  light  in  the  dielectric. 

It  is  because  the  coefficient  k  is  not  an  arbitrary  factor, 
but  an  actual  physical  quantity,  that  we  have  retained  it 
in  our  formulae.  The  importance  of  this  coefficient  will 
be  seen  in  the  comparison  of  electrical  units. 

105.  Role  of  the  Dielectrics.  Displacement. — What  we 
have  just  seen  shows  the  importance  of  the  role  of  the  di- 
electric in  electrostatic  actions.  We  present  here  an  ex- 
periment which  leads  to  the  consideration  of  the  phenome- 
non of  condensation  (or  accumulation)  in  a  new  light. 

A  condenser  formed  of  three  detachable  parts,  a  glass 
jar  or  plate  and  two  metallic  plates,  is  charged  in  the  ordi- 


PROPERTIES  OF  ELECTRIFIED  BODIES.  143 

nary  way.  The  three  parts  are  then  separated  and  the 
metal  plates  connected  to  earth  ;  then  the  condenser  is  put 
together  again,  and  it  can  be  discharged  as  if  it  had  remained 
undisturbed. 

This  experiment  sho,ws  that  ^the  charge  is,  at  least  in  part, 
on  the  dielectric. 

Faraday  has  deduced  an  ingenious  hypothesis  from  this 
fact.  According  to  him,  dielectrics  present  a  polarization 
which  recalls  that  which  we  have  met  with  in  the  phenomena 
of  magnetization.  This  polarization  takes  place  along  the 
lines  and  tubes  of  force  which  join  the  plates  of  condensers 
and  electrified  bodies  in  general. 

Take  the  case  of  an  elementary  tube  between  two  con- 
ductors A  and  B  joining  corresponding  elements  ds1  ,  dj, 
charged  with  densities  crl  ,  0*2  ,  §  88. 

This  tube  can  be  subdivided  by  equipotential  sections 
into  elements  of  volume  presenting  equal  and  contrary 
charges  on  their  opposing  ends,  thus  balancing  each  other. 
At  the  extremities  of  the  tube  exist  quantities  of  free 
electricity, 

+  q  =  o-.ds,  , 
-  q  =  a,ds,  , 

which  charge  the  conductors,  these  latter  having  no  other 
function  than  to  limit  the  dielectric. 

In  this  hypothesis  the  product  ads  =  CTI^SI  is  constant 
along  the  tube  ;  but  at  the  surface  of  the  conductors,  §  86, 
we  have 


and  as  the  product  3C/ta  =•  3tlds1  is  constant  in  the  tube, 
§21,  we  conclude  that  in  any  point  of  the  field  the  quantity 
of  electricity  displaced  per  unit  equipotential  surface  is 


144  ELECTRICITY. 

equal  to  the  intensity  of  the  field  at  this  point  divided  by 
the  factor  ^nk  : 


Maxwell  calls  this  expression  the  displacement  of  electricity 
in  the  dielectric. 

The  analogy  which  exists  between  electric  induction  thus 
interpreted  and  magnetic  induction  will  be  recognized  : 
electric  displacement  corresponding  to  intensity  of  magneti- 
zation, and  the  specific  inductive  capacity  to  the  coefficient 
of  magnetization  (magnetic  susceptibility}. 

This  point  of  view  gives  us  an  explanation  of  the  energy 
of  electrified  conductors  ;  we  conceive,  namely,  that  there 
results  from  the  polarization  of  the  dielectric  molecules  a 
state  of  tension  along  the  lines  of  force,  in  virtue  of  which 
these  lines  tend  to  shorten  themselves  by  causing  the 
approach  of  the  conductors  which  limit  the  dielectric. 
This  tension,  which  is  comparable  with  that  of  an  elastic 
body,  assimilates  the  energy  of  the  dielectric  to  that  of  a 
spring. 

It  can  be  readily  verified  that,  if  the  electric  field  is 
uniform  between  the  plates  of  a  condenser,  such  as  a  guard- 
ring  condenser,  the  total  energy  is  expressed  by  the  volume 

3Ca 
of  the  dielectric  multiplied  by  -—-j.,  3C,  being  the  intensity  of 

the  field. 

In  fact,  keeping  the  previous  notation, 

W=teU=i<rsX  Wed  =  43e/r  X  volume. 
Now 


whence 

W=  volume  X 


PR OPER  TIES  OF  ELECTRIFIED  B OD IES.  1 4 5 

In  the  case  of  a  uniform  field  the  energy  of  the  dielectric 
per  unit  volume  is  therefore  expressed  by 


Maxwell  has  shown"  that  this  holds  in  the  case  of  any 
field  whatever,  and  that  the  law  of  the  attractions  and 
repulsions  of  electrified  bodies  can  be  established  by 

t"*C2 

admitting,  along  the  lines  of  force,  a  tension  of    *'  e       per 

ortfe 

unit  of  surface  normal  to  the  field.* 

106.  Residual  Charge  of  a  Condenser. — The  polarized 
condition  of  a  solid  dielectric  does  not  always  cease  after 
the  discharge  of  the  plates. 

Suppose  we  charge  a  Leyden  jar,  the  inner  coating  of 
which  is  connected  with  the  needle  of  a  quadrant-electrom- 
eter, while  the  outer  coating  is  connected  with  the  earth. 
At  the  moment  of  discharging  the  jar  the  needle  returns  to 
zero,  but  gradually  reassumes  a  deviation  but  slightly  differ- 
ent from  the  first.  The  discharge  may  be  repeated  a  certain 
number  of  times  before  the  needle  finally  rests  at  zero. 

This  phenomenon,  which  presents  some  analogy  with  the 
remanent  magnetism  of  a  bar  of  steel,  is  particularly  ob- 
servable in  the  case  of  dielectrics  which  are  not  pure,  such 
as  glass,  gutta-percha,  caoutchouc  ;  sulphur  and  paraffine 
exhibit  this  phenomenon  to  a  much  smaller  extent. 

The  electrification  of  the  plates  after  a  first  discharge  is 
called  the  residual  charge.  This  varies  with  the  duration  of 
the  electrification  of  the  condenser,  while  the  effective  charge 
that  can  be  obtained  at  the  moment  of  the  first  contact 
between  the  plates  is  independent  of  this  duration.  For 
this  reason  it  has  been  agreed  to  call  the  ratio  of  the  effective 

*  Maxwell,  Electricity  and  Magnetism,  Vol.  I,  Chap.  V,  etc. 


1 46  ELECTRIC1T  Y. 

charge  to  the  difference  of  potential   of  the   plates  the  ca- 
pacity of  the  condenser. 

From  the  preceding  considerations  it  follows  that,  for  the 
same  difference  of  potential  of  the  plates,  the  charge  of  a 
condenser  is  greater  with  decreasing  than  increasing  differ- 
ences of  potential.  The  electrization  of  a  dielectric  placed 
in  an  electrical  field  of  varying  intensity  presents,  it  will  be 
seen,  a  true  parallel  with  the  magnetization  of  iron  intro- 
duced into  a  field  of  varying  intensity,  which  has  resulted  in 
the  name  of  dielectric  hysteresis  being  given  to  the  phenom- 
enon here  considered.  In  both  cases  the  cyclical  variations 
of  the  intensity  of  the  field  causes  heating. 

107.  Electromotive  Force  of  Contact.  Distinction 
between  Electromotive  Force  and  Difference  of  Poten- 
tial.— Volta  discovered  that  two  metals,  e.g.,  zinc  and 
copper,  assume  a  definite  difference  of  potential  on  contact. 

Following  Lord  Kelvin's  experiment,  let  us  make  a  disk, 
one  half  being  of  copper  and  the  other  of  zinc,  and  hang  a 
slender  sheet  of  metal  over  the  line  of  contact.  When  this 
sheet  is  positively  electrified,  it  is  displaced  towards  the 
copper ;  electrified  negatively,  it  is  attracted  towards  the 
zinc. 

We  might  also  make  the  electrometer  of  §  79  with  two 
quadrants,  A,  B,  of  zinc,  and  the  other  two,  C,D,  of  copper 
On  connecting  the  zinc  and  copper  quadrants  by  copper 
wire  we  will  get  a  deviation  of  the  needle  towards  the  zinc 
or  towards  the  copper,  according  as  its  charge  is  negative 
or  positive. 

This  experiment  shows  that  upon  contact  the  zinc  is 
positively  charged,  the  copper  negatively.  In  virtue  of 
these  charges  the  two  metals  take  opposite  potentials,  whose 
difference  is  called  the  electromotive  force  of  contact. 

Similar  electromotive  forces  probably  arise  on  the  con- 


PROPERTIES  OF  ELECTRIFIED  BODIES.  147 

tact  of  all  dissimilar  bodies.  Helmholtz  supposes  them  to 
be  due  to  the  different  specific  attractions  of  the  bodies  for 
electricity.  Those  whose  attraction  is  greater  take  elec- 
tricity in  excess,  that  is,  are  positively  charged.  At  the 
surface  of  separation  are  produced  two  opposite  layers  of 
electricity,  occasioning  the  observed  difference  of  potential. 

This  hypothesis  accounts  for  the  fact,  shown  by  experi- 
ment, that  the  same  body  is  capable  of  taking  opposite 
charges,  according  to  the  nature  of  the  bodies  placed  in 
contact  with  it.  Copper,  for  example,  is  negative  with 
regard  to  zinc  and  positive  with  regard  to  silver. 

In  a  general  way,  the  name  electromotive  force  is  reserved 
for  every  cause  of  displacement  of  electricity ;  a  difference 
of  potential  is  the  effect  of  an  electromotive  force.  This 
distinction  is  analogous  to  that  between  the  pressure  exer- 
cised on  a  liquid  and  the  difference  of  level  which  results 
therefrom  in  a  manometer. 

The  electromotive  force  of  contact  gives  an  explanation 
of  the  development  of  electricity  by  friction,  which,  by  this 
hypothesis,  has  no  other  effect  than  to  render  the  contact  of 
the  rubbed  bodies  more  thorough. 

The  high  potential  acquired  by  bodies  when  rubbed  is 
explained  by  the  separation  of  the  opposite  charges. 
Suppose,  for  example,  two  discs,  one  of  zinc,  the  other  of 
copper,  in  contact.  The  potentials  assumed  by  these  two 
metals  depend  on  their  charges  and  relative  distances ; 
e.g.,  for  a  point  on  the  zinc  disc  the  potential,  which  is 
constant  in  the  whole  homogeneous  mass,  is  given  by  the 
sum  of  the  ratios  of  the  positive  charges  of  the  disc  to  their 
distances  from  this  point,  less  the  sum  of  the  ratios  of  the 
negative  charges,  distributed  over  the  adjoining  disc,  to 
their  corresponding  distances. 

If  the  two  discs  are  moved  apart,  the  subtractive  term 
diminishes  very  rapidly,  so  that  the  positive  potential  of  the 


1  48  ELECTRICIT  Y. 

zinc  becomes  considerable,  as  does  also  the  negative  poten 
tial  of  the  copper. 

The  increase  in  potential  energy, 


which  results  is  equal  to  the  work  expended  in  separating 
the  opposite  charges. 

ELECTRIC   DISCHARGES  AND   CURRENTS. 

So  far  we  have  been  studying  the  properties  of  conduct- 
ors charged  with  layers  of  electricity  retained  by  dielectrics 
and  supposed  to  be  in  equilibrium.  Let  us  now  take  up  the 
case  of  rupture  of  equilibrium  and  the  manifestations  result- 
ing from  that  rupture. 

108.  Convective  Discharge.  —  The  outside  coating  of  a 
charged  Leyden  jar,  Fig.  52,  is  connected  with  a  metallic 
knob  bf,  in  proximity  to  the  knob  b  of  the  inner  coating.  A 
pith-ball  s,  being  hung  by  a  silk  thread  between  the  two 
knobs,  is  electrified  by  influence  and  attracted  by  the  nearest 


FIG.  52. 

knob,  on  coming  into  contact  with  which  it  takes  a  charge 
of  the  same  sign.  It  then  swings  to  the  opposite  knob  and 
continues  to  oscillate  from  one  knob  to  the  other,  carrying 
small  electric  charges,  until  the  opposite  electricities  of  the 
coatings  are  neutralized. 


ELECTRIC  DISCHARGES  AND    CURRENTS.  149 

This  phenomenon  may  be  interpreted  as  a  transporting 
of  positive  electricity  from  b  to  b' ',  and  of  negative  electricity 
in  the  opposite  direction.  On  the  hypothesis  of  a  single 
fluid,  the  whole  transportation  .is  from  b  to  b'. 
•  The  potential  energy  of  the^  jar,  %  2qU,  is  gradually  re- 
duced to  zero  ;  it  is  used  up  in  the  blows  of  the  swinging 
ball  against  the  knobs  b  and  b'  and  the  friction  of  the  ball 
against  the  air;  a  transformation  of  electric  energy  into 
heat  therefore  takes  place. 

This  discharge  of  the  jar  is  called  connective,  because  the 
displacement  of  electricity  is  connected  with  the  movement 
of  the  body  which  conveys  it. 

109.  Conductive   Discharge.    Electric    Current. — The 

contrary  electricities  which  charge  the  coatings  are  made  to 
recombine  directly,  if  they  are  joined  together  by  a  conduc- 
tive wire.  The  electrification  is  then  observed  to  disappear 
instantaneously  without  displacement  of  the  conductor,  but 
we  can  show  that  also  in  this  case  there  is  a  transformation 
of  electric  energy  into  heat  in  the  conductor. 


FIG.  53. 

For  this  purpose  Riess*  thermometer  is  used,  which  con- 
sists of  a  glass  bulb  A  traversed  by  a  platinum  spiral  s. 
The  bulb  communicates  with  an  inclined  glass  tube  E,  ter- 
minating in  a  vertical  tube  T^which  contains  a  colored  liquid. 
When  a  discharge  is  passed  through  the  platinum  spiral,  the 
heat  developed  in  it  is  communicated  to  the  air  in  the  bulb 


150  ELECTRICITY. 

and  causes  a  sudden  displacement  of  the  liquid  in  the  in- 
clined tube  ;  the  expansion  of  the  air  enables  us  to  calculate 
the  heat  given  off. 

If  we  neglect  the  loss  by  radiation,  then,  denoting  by  c 
the  specific  heat  of  platinum,  m  the  mass  of  the  wire,  c' ,  m' 
the  same  terms  for  the  air  in  the  bulb,  d  the  alteration  of 
level  of  the  liquid,  #and  #'the  initial  and  final  temperatures, 
the  heat  developed  is 

W  =  (me  +  m'cf}(df-e)  =  ad, 

a  being  a  numerical  coefficient. 

The  mechanism  of  the  conductive  discharge  is  explained, 
like  that  of  the  convective,  by  supposing  that  the  molecules 
of  the  conductor  play  the  same  part  as  the  pith-ball ;  ex- 
changes of  electricity  take  place  from  molecule  to  molecule 
by  the  displacement  of  these  latter.  The  increase  in  mo- 
lecular activity  corresponds  to  the  observed  development 
of  heat. 

It  must  not,  however,  be  forgotten  that  before  the  intro- 
duction of  the  conductor  between  the  coating  the  seat  of 
the  energy  was  in  the  dielectric,  §  105.  The  transfer  of 
energy  from  the  dielectric  to  the  conductor  implies  that  at 
the  moment  of  establishing  the  metallic  connection  the  lines 
of  electric  force  crowd  together  into  the  conductor,  in  which 
a  polarization  is  thus  produced,  followed  by  transfers  of  elec- 
tricity due  to  the  molecular  agitation. 

The  mechanism  of  this  transfer  still,  however,  remains 
to  be  explained.  We  shall  see,  when  studying  variable  cur- 
rents, that  the  development  of  heat  is  first  produced  in  the 
outer  parts  of  the  conductor,  and  that  when  the  phenomenon 
is  sufficiently  rapid  it  cannot  reach  the  middle  of  the 
conductor. 

Another  way  of  looking  at  the  matter,  less  exact,  perhaps, 


ELECTRIC  DISCHARGES  AND    CURRENTS.  1$! 

but  more  convenient,  suggests  the  ideas  of  hydrodynamic 
phenomena,  consists  in  supposing  a  simple  transportation  or 
current  of  electric  fluid  in  the  conductor  joining  the  coat- 
ings taking  place  from  the  ^positive  knob  to  the  negative 
one.  The  heat  produced  in  tlj£  conductor  is  then  explained 
in  the  same  way  as  that  caused  by  a  ponderable  fluid  rub- 
bing against  the  sides  of  a  pipe  ;  the  movement  of  the  elec- 
tric fluid  is  communicated  to  the  molecules  of  the  conductor 
which  it  thus  heats. 

The  quantity  of  electricity  which  passes  per  second  at 
any  given  time  across  the  section  of  the  conductor  is  called 
the  strength  of  current  (or  simply  the  current).  It  has  been 
agreed  to  assume  that  the  current  is  directed  from  the  posi- 
tive to  the  negative  plate. 

1 10.  Disruptive  Discharge.  Electric  Spark  and 
Brush.  Their  Effects. — If  we  adopt  Faraday's  and  Max- 
well's view,  we  may  suppose  that,  upon  gradually  increasing 
the  difference  of  potential  of  the  plates  of  a  condenser,  there 
comes  a  time  when  the  tension  of  the  dielectric  is  no  longer 
balanced  by  its  tenacity,  so  that  a  rupture  is  produced  ;  this 
is,  in  fact,  shown  by  experiment.  The  dielectric  of  a  con- 
denser is  pierced  when  the  tension  exceeds  a  certain  limit, 
and  there  is  then  a  sudden  return  to  the  neutral  state,  while 
at  the  same  time  a  spark  is  observed  to  pass  between  the 
plates  ;  this  discharge  is  called  disruptive. 

In  gases  the  sparking  takes  place  at  a  much  lower  ten- 
sion than  in  solid  dielectrics.  The  length  of  the  spark  for  a 
given  tension  naturally  depends  on  the  shape  of  the  con- 
ductors used,  since  the  electrostatic  pressure  varies  as  the 
square  of  the  density  on  the  conductors,  §  87.  It  varies 
also  with  the  pressure  of  the  gas;  compression  diminishes 
the  explosive  distance,  while  rarefaction  increases  the  length 
of  the  spark  up  to  a  certain  limit,  beyond  which  the  de- 


152  ELECTRICITY. 

crease  is  rapid.  In  a  vacuum  the  disruptive  discharge  com- 
pletely ceases. 

Matter  is  therefore  necessary  for  the  transfer  of  electric- 
ity, which  seems  to  show  that  it  is  the  material  molecules 
which  serve  as  its  vehicle,  as  in  the  case  of  convective  dis- 
charge. 

In  a  gas  under  constant  pressure  and  with  pointed  elec- 
trodes the  length  of  the  spark  grows  much  more  rapidly  than 
the  potential  difference  of  the  conductors  ;  a  fact  which  leads 
us  to  think  that  the  large  sparks  due  to  storm-clouds  are 
not  caused  by  potentials  out  of  proportion  to  those  of  our 
electrostatic  machines. 


FIG.  54. 

The  spark  assumes  very  diverse  aspects,  according  to  the 
nature  of  the  gas  in  which  it  is  produced  and  the  form  of 
the  conductors.  Sometimes  it  takes  the  form  of  a  straight, 
curved,  or  zigzag  incandescent  flash,  and  produces  a  violent 
detonation.  A  photograph  of  this  spark  shows  that  it  is 
composed  of  a  great  number  of  interlaced  luminous  lines, 
marked  with  points  more  brilliant  than  the  rest.  At  other 
times  the  spark  seems  to  branch  out,  as  shown  in  the  bottom 
figure. 


ELECTRIC  DISCHARGES  AND    CURRENTS.  153 

The  duration  of  the  spark  from  our  machines  is  very 
short,  for  the  illumination  which  it  produces  enables  us  to 
photograph  objects  in  very  rapid  motion,  such  as  projectiles; 
but  the  lightning-flashes  wrych  occur  during  storms  often 
have  sufficient  duration  to  allo^v  the  shaking  movement  of 
leaves  of  trees  to  be  distinguished  at  night. 

Spectroscopic  examination  shows  that  the  spark  gives 
the  lines  of  the  metals  composing  the  conductors,  and  of 
the  gases  in  which  it  is  produced,  which  gives  us  reason  to 
attribute  the  light  of  the  spark  to  the  incandescence  of  par- 
ticles violently  torn  away  and  volatilized.  When  the  con- 
ductors are  of  different  substances,  it  is  observed  that  mat- 
ter is  transported  from  one  to  the  other  in  both  directions; 
a  fact  which  gives  evidence  of  an  oscillating  movement  in 
the  discharge-current.  Further  on  will  be  seen  a  confirma- 
tion of  this  deduction. 

When  one  of  the  conductors  has  sharp  ridges  or  points, 
the  discharge  is  continuous  and  takes  the  form  of  a  phos- 
phorescent, violet-tinted  brush,  accompanied  by  a  peculiar 
hissing  noise  and  a  current  of  air  called  an  electric  breeze. 
Faraday  has  explained  this  current  by  the  displacement  of 
gaseous  molecules,  which  are  electrified  by  contact  with  the 
point  and  are  then  repulsed  towards  the  conductors  in  the 
vicinity,  which  shows  clearly  the  convective  nature  of  the 
phenomenon.  In  rarefied  gases  the  electric  discharge  is 
made  evident  by  a  glow  attributed  to  the  impact  of  the 
molecules  which  have  been  set  in  motion.  When  the  vac- 
uum is  carried  beyond  a  certain  point,  the  glow  only  ap- 
pears on  the  internal  surface  of  the  vessel  containing  the 
rarefied  gas,  which  seems  due  to  the  fact  that  the  gaseous 
molecules  no  longer  impinge  during  their  passage  in  the 
vessel,  but  strike  against  its  walls.  By  modifying  the  nature 
of  the  glass  of  which  the  vessel  is  made  we  can  give  different 
colors  to  the  glow. 


I  5 4  ELECTRICIT  Y. 

The  molecular  agitation  caused  by  the  electric  spark  is 
especially  favorable  to  chemical  reactions;  the  combination 
of  oxygen  with  hydrogen,  the  production  of  ozone,  the  odor 
of  which  accompanies  the  spark,  are  very  well  known  phe- 
nomena. Ozone,  which  is  used  at  the  present  time  for  the 
purification  of  alcohols  and  wines,  is  obtained  by  causing  a 
current  of  air  dried  by  sulphuric  acid  to  pass  through  a 
space  in  which  a  glow-discharge  is  maintained. 

Another  property,  shown  by  Prof.  O.  Lodge,  is  the  con- 
densation of  vapors,  smoke,  or  dust  in  a  medium  traversed 
by  an  electric  spark.  If  electric  discharges  are  produced  in 
an  atmosphere  charged  with  particles  in  suspension,  there 
arise  vortices,  followed  by  the  precipitation  of  the  particles 
on  the  surrounding  walls.  It  has  been  proposed  to  apply 
this  property  to  the  condensation  of  the  metallic  dust  in 
the  flues  of  lead  and  zinc  retorts,  and  even  to  the  condensa- 
tion of  the  heavy  fogs  which  are  formed  in  many  cities. 

LAWS   OF   THE   ELECTRIC   CURRENT. 

III.  Having  now  passed  in  review  the  different  aspects 
of  the  discharge  of  electrified  bodies,  the  law  of  the  move- 
ment of  electricity  in  conducting  bodies  will  next  be  con- 
sidered. 

When  the  plates  of  a  condenser  are  connected  by  a  wire, 
the  discharge  is  so  rapid  that  the  course  of  the  phenomenon 
cannot  be  followed  by  any  known  experimental  method. 
Its  analysis  is  likewise  extremely  difficult  in  consequence 
of  the  effects  of  electromagnetic  induction  which  are  pro- 
duced in  the  wire,  as  we  shall  see  further  on. 

The  complexity  of  the  case  of  an  electric  flux  in  a  con- 
ductor whose  ends  are  at  a  variable  difference  of  potential 
leads  us  to  investigate  first  of  all  the  case  of  a  conductor 
whose  ends  are  maintained  at  a  constant  difference  of  po- 
tential. 


LAWS   OF   THE  ELECTRIC   CURRENT.  155 

Let  us  first  examine    the  means  for   reaching  such  a  re- 

sult. 

112.  Law  of  Successive  Contacts.  —  We  have  seen, 
§  107,  that  two  heterogenedus  bodies  furnish,  on  coming 
into  contact,  a  difference  of  potential  called  electromotive 
force  of  contact,  which  is  shown  by  means  of  an  electrometer. 

When  a  number  of  conductors,  A,  B,  C,  D,  at  the  same 
temperature,  form  an  open  chain,  the  electromotive  forces 
between  the  bodies  in  contact  are  added  together  or  sub- 
tracted, according  to  their  direction,  and  their  algebraic  sum 
constitutes  the  difference  of  potential  between  the  extremi- 
ties of  the  series.  If  these  extremities  be  joined  together 
so  as  to  form  a  closed  chain  of  conductors,  the  algebraic  sum 
of  the  electromotive  forces  must  be  zero,  or  else  a  perma- 
nent current  of  electricity  would  be  produced  in  the  circuit, 
which  would  occur  without  any  expenditure  of  energy. 

Denoting  by  A  \  B,  B\  C,  C\D  the  successive  differences 
of  potential,  we  must  have 


whence 

A  |  B  +  B\  C+  C\  D  =  -  D\A  =  A  \  D. 

We  see,  then,  that  in  a  series  of  conductors  connected  one  to 
another  the  difference  of  potential  of  the  end  conductors  is 
the  same  as  if  they  were  directly  connected. 

The  solder  which  joins  two  metals  is  therefore  without 
effect  upon  their  difference  of  potential. 

113.  Thermal  and  Chemical  Electromotive  Forces. 
Means  of  Keeping  Up  a  Constant  Difference  of  Poten- 
tial in  a  Conductor.  —  The  preceding  law  fails  in  two  cases: 

i.  When  the  successive  contacts  are  kept  at  different 
temperatures. 


1 56  ELECTRICIT  Y. 

2.  When  the  successive  conductors  act  chemically  upon 
each  other. 

The  first  case  was  discovered  by  Seebeck. 

Suppose  a  copper  wire  forming  a  closed  circuit  with  a 
zinc  wire.  When  one  of  the  junctions  is  heated,  a  flux  or 
current  of  electricity  is  produced,  flowing  from  the  copper 
to  the  zinc  across  the  hot  junction.  With  the  aid  of  a  suffi- 
ciently delicate  electrometer  it  will  be  found  that  in  each  of 
the  wires  the  extremities  are  at  a  constant  difference  of  poten- 
tial, the  temperature  of  these  extremities  remaining  invariable. 

The  discovery  of  the  second  case  is  due  to  Volta. 

Form  a  circuit  by  fastening  a  zinc  and  a  copper  wire  to- 
gether and  plunging  their  free  ends  into  dilute  sulphuric 
acid,  which  closes  the  circuit.  An  electric  current  is  pro- 
duced in  the  conductors,  for  the  electrometer  shows,  in  the 
copper  wire,  potentials  decreasing  towards  the  zinc,  and,  in 
the  zinc  wire,  potentials  decreasing  towards  the  acid. 

These  two  experiments,  which  constitute  the  foundation 
of  thermo-electric  and  hydro-electric  batteries  (to  which  we 
will  return  further  on),  furnish  the  means  of  maintaining  a 
constant  difference  of  potential  between  the  ends  of  a 
conductive  wire  placed  in  the  circuit.  We  will  call  these 
electromotive  forces  which  form  an  exception  to  the  law  of 
successive  contacts  by  the  name  of  thermal  or  chemical 
electromotive  forces. 

114.  Ohm's  Law. — Suppose  a  homogeneous  conductor, 
of  any  shape,  two  points  of  which,  A  and  B,  are  maintained, 
as  shown  above,  at  different  and  constant  potentials. 

An  electric  field  is  formed  in  the  conductor  whose  lines 
of  force  go  from  the  point  where  the  potential  is  highest,  A, 
for  example,  towards  the  point  with  the  lowest  potential. 

As  the  electricity  can  move  along  the  lines  of  force,  since 
the  field  is  a  conductor,  it  can  be  naturally  admitted  that  in 


LAWS   Of    THE   ELECTS  1C   CURRENT.  !$? 

each  tube  of  force  there  takes  place  a  transportation  of 
electricity  proportional;  to  the  corresponding  flux  of  force. 

If,  in  an  element  of  equipotential  surface,  the  flux  of 
force  is  Hds,  the  flux  of  electricity  per  second  will  be  repre- 
sented by  yHds,  y  being  a  constant  for  an  isotropic  con- 
ductor. 

The  total  flux  of  electricity  per  second  across  an  equi- 
potential section  of  the  conductor  will  be  the  sum  of  the 
elementary  fluxes,  such  as  yHds. 


This  quantity  is  called  the  strength  of  current.  The 
direction  of  the  displacement  of  positive  electricity  is  taken 
as  the  direction  of  the  current.  The  factor  y  is  called  the 
specific  conductivity. 

115.  Case    of  a    Conductor  of   Constant    Section. — 

The   ends  of  a  homogeneous  cylindrical    conductor   being 
maintained  at  constant  potentials  Ul  >  C7t,  the  lines  of  force 
are  parallel  to  the  axis  of  the  cylinder. 
At  any  point      . 

AU 
"  d/' 

d/  denoting  an  element  parallel  to  the  axis. 

As  H  is  constant  all  along  the  cylinder,  we  have 

/»/  />£/, 

H       d/=-/ 

t/o  l/t/! 

whence 

Hl=U,-U,. 

Consequently  the  current 


I  5  8  ELECTRICIT  Y. 

where  the  sign    /   must  be  extended  to  every  section  of  the 
conductor  ;  consequently 


r_ 


ys 


The  same  method  of  reasoning  can  be  applied  to  a  con- 
ductor of  any  shape  and  section  whatever,  provided  that  the 
latter  be  constant  throughout  the  conductor. 

The   ratio  —  is  called  the  resistance  of  the  conductor  ;  —  , 
ys  y 

which  measures  the  resistance  of  a  conductor  having  unit 
length  and  unit  section,  is  the  specific  resistance  or  resistivity. 

It  is  seen  that  the  resistivity  is  the  inverse  of  the  con- 
ductivity, just  as  the  resistance  of  a  conductor  is  the  inverse 
of  its  conductance. 

The  above  equation  was  discovered  by  Ohm.  It  is 
enunciated  as  follows  : 

The  strength  of  the  current  between  two  points  of  a  con- 
ductor of  any  form,  but  of  constant  section,  is  directly  propor- 
tional to  the  difference  of  potential  and  inversely  proportional 
to  the  resistance  between  those  points. 

By  virtue  of  the  definition  given  of  strength  of  current 
the  quantity  of  electricity  which  passes  across  a  section  of 
the  conductor  in  a  time  t  is 


This  evident  equation  is  sometimes  called  Faraday's  law. 

116.  Graphic  Representation  of  Ohm's  Law.—  Ohm's 
law  proves  that  in  a  conductor  of  constant  section  the  poten- 
tial falls  uniformly,  for  the  relation 


ys 


LAWS   OF   THE  ELECTRIC   CURRENT.  1 59 

shows  that  the  variation  of  the  potential  is  proportional  to 
the  variation  of  the  length. 

Mark  off  on  a  horizontal  line  Ox  a  length  OA,  measuring, 
on  a  given  scale,  the  resistance-^  of  a  conductor. 


FIG.  55- 

On  the  perpendiculars  at  the  extremities  of  OA  mark  off 
OB  and  AC  to  represent  the  potentials  at  the  two  ends  of 
the  conductor;  then  the  ordinates  of  the  straight  line  BC 
represent  the  potentials  of  the  intermediate  points  of  the 
conductor. 

The  strength  of  the  current  is  measured  by  the  tangent 
of  the  angle  between  BC  and  OA,  for  we  have 

OB  -AC       U-U 


117.  Variable  Period  of  the  Current.  —  Suppose  a  con- 
ductor traversed  by  a  current  and  enveloped  in  a  dielec- 
tric sheathing,  which  latter  is  surrounded  by  a  conductive 
layer:  this  is  the  case  with  an  insulated  conductor  under 
water.  If  the  outside  conductor  is  connected  with  the  earth, 
its  potential  is  zero,  and  an  electric  field  is  formed  in  the  di- 
electric, the  lines  of  force  passing  from  the  inside  conductor 
to  the  outside  or  vice  versa,  according  as  the  potential  of  the 
inner  conductor  is  greater  or  less  than  zero. 

These  lines  of  force  join  points  carrying  opposite  charges 
(§  88),  so  that  the  adjoining  surfaces  of  the  dielectric  as- 
sume contrary  electrifications,  as  in  the  case  of  a  condenser. 


1  60  ELECTRICIT  Y. 

The  charge  per  unit  length  will,  however,  vary  at  different 
points  of  the  conductor.  If,  for  example,  the  potential  de- 
creases uniformly  from  U  to  o  in  the  conductor,  the  charge 
will  also  decrease  regularly  from  one  extremity  to  the  other, 
and  the  total  charge  of  the  conductor  will  be  equal  to  the 
capacity  of  the  condenser  multiplied  by  the  mean  difference 
of  potential  of  the  two  plates. 

To  explain  the  phenomenon  of  condensation  we  must 
admit  that  at  the  moment  when  a  difference  of  potential 
occurs  in  the  conductor  a  momentary  flux  of  electricity 
traverses  its  surface  to  produce  the  surface-distribution. 
This  variable  period  is  followed  by  the  permanent  period, 
during  which  the  electric  flux  is  constant  and  passes  in  its 
entirety  parallel  to  the  walls  of  the  conductor. 

118.  Application  of  Ohm's  Law  to  the  Variable  Period 
of  the  Current  in  but  Slightly  Conductive  Bodies.  — 

When  the  plates  of  a  condenser  are  joined  by  an  imperfect 
insulator,  such  as  caoutchouc  or  gutta-percha,  it  is  observed 
that  they  lose  their  electrification  little  by  little.  This  loss 
can  be  attributed  to  a  certain  conduction  through  the  given 
substance,  and  Ohm's  law  can  be  applied  to  this  special  case 
of  a  variable  period.  Let  C  be  the  capacity  of  the  con- 
denser, R  its  resistance,  ^the  difference  of  potential  between 
the  plates. 

At  any  moment  the  current  /  is  the  ratio  of  the  potential 
difference  to  the  resistance, 


The  quantity  of  electricity  which  passes  during  a  time 
d/is 

-  d?  =  7d/ 
at  that  instant  of  time. 


LAWS  OF  THE  ELECTRIC  CURRENT.  l6l 

But 

q  =  CU.     (§82.) 
Consequently 

d?= 
whence 


The  time  that  the  difference  of  potential   takes  to  pass 
from  £7,  to  £/3  is 

/",, 

Jo     dt  =  - 

or 


119.  Application  of  Ohm's  Law  to  the  Case  of  a 
Heterogeneous  Circuit.  —  Suppose  two  conductors  ab,  be, 
of  resistances  Rl  and  R^  ,  in  contact  at  the  point  b,  and  whose 
free  ends  a  and  c  are  maintained,  by  any  means  whatever, 
at  constant  potentials  Ul  and  U^  .  A  current  will  proceed 
from  the  point  of  higher  potential  to  that  of  lower  potential. 

Suppose 

U,>  U,. 

At  the  point  of  contact  b  an  electromotive  force  E  is 
produced,  being  the  difference  of  the  potentials  £/"/,  Ut',  of 
the  two  sides  of  the  surface  of  separation. 

Suppose 

v;  >  u,\ 

we  have 

E=Ul'  -  £/,'. 

Let  us  apply  Ohm's  law  to  the  two  portions  ab,  be, 
observing  that,  as  the  electricity  can  be  neither  accumu- 
lated at  nor  abstracted  from  the  point  b  during  the  perma- 


1  62  ELECTRICIT  Y. 

nent  stage,  the  current  must  necessarily  be  the  same  in  the 
two  conductors. 

We  shall  therefore  have 


/?,  A. 

U,-U,-E 

R>+R* 

In  the  case  where 

u,f  >  u 

T_V,-U, 


the  electromotive  force  E  being  taken  wjth  the  -f-  or  —  sign 
according  as  it  gives  an  increase  or  decrease  of  potential  in 
the  direction  of  the  current,  that  is  to  say,  according  as  it 
tends  to  increase  or  diminish  the  current. 

By  extension,  if  there  are  several  conductors  in  contact, 
El  ,  E9,  E3  .  .  .  being  the  electromotive  forces,  R1  ,  R,,  Rt 
.  .  .  the  resistances,  we  have 

r  &> 


the  signs  of  the  electromotive  forces  being  obtained  by  the 
preceding  rule. 

If  we  connect  the  end  conductors  directly,  we  will  have 


We  know  that  2£  is  zero,  unless  there  is  a  difference  of 
temperature  or  chemical  action  at  the  points  of  junction 

(§"  3). 

120.  Graphic  Representation.  —  Take  the  case  of  three 
conductors  ;  mark  off  successively  on  the  axis  of  abscissae 
lengths  proportional  to  their  resistances  Riy  R^,  R3. 


LAWS   OF   THE  ELECTRIC  CURRENT. 


163 


Let   aa!  =  Ul  (Fig.   56).       From  the  point  a'  draw  the 
right  line  a'b'  inclined  at  an  angle  a,  such  that 

=  I. 


tan 


If  the  electromotive  force  ofc  contact  El  is  positive,  meas- 
ure it  off  along  b'b"  and  from  b"  draw  b"c  parallel  to  a'b'. 


~ — 'd 


FIG.  56. 


The  electromotive  force  E9 ,  supposed  to  be  negative,  is 
marked  off  on  c'c"  and  the  new  line  c"d!  will  cut  the  poten- 
tial-line dd'  at  the  point  d' '. 

121.  Kirchhoffs  Laws. — This  name  is  given  to  two 
laws,  one  of  which  is  evident  when  we  compare  the  electric 
current  with  a  fluid  current,  the  other  deduced  from  Ohm's 
law ;  the  two  together  enabling  us  to  solve  the  problem  of 
even  the  most  complicated  electric  circuits. 

First  Law. —  When  any  number  of  conductors  meet  in  a 
point,  the  algebraic  sum  of  the  currents  at  that  point  is  zero. 

This  rule  simply  expresses  the  fact  electricity  can  be 
neither  accumulated  nor  subtracted  at  the  meeting-point  of 
conductors.  The  currents  are  considered  as  of  opposite 
sign,  according  as  they  flow  into  or  away  from  the  point. 

Second  Law. — In  every  closed  circuit,  the  algebraic  sum  of 
the  electromotive  forces  equals  the  algebraic,  sum  of  the  products 
of  the  currents  by  the  resistances  of  the  conductors. 

Let  abed  be  a  circuit  in  a  network  of  conductors. 

Let  us  denote  by  i , ,  za,  i 3 ,  i<  the    currents  whose  direc- 


164 


ELECTRICITY. 


tion  is  shown  by  the  arrows,  and  by  rl  ,  r^ ,  ra ,  rt  the  resist- 
ances, and  by  *, ,  *a ,  *, ,  ^4  the  electromotive  forces  shown 
by  the  unequal  parallel  strokes.  The  -j-  sign  shows  the 


FIG.  57. 

direction  in  which  each  electromotive  force  tends  to  produce 
an  increase  of  potential. 

Let  ul ,  &a ,  ua ,  &4  be  the  potentials  at  the  points  a,  bt  c9  d. 

We  shall  then  have,  by  §  113, 


whence 


or 


The  signs  which  should  be  given  to  the  currents  and 
electromotive  forces  are  easily  determined,  viz.,  follow  the 
circuit  round  in  the  direction  of  movement  of  the  hands  of 
a  watch ;  give  the  sign  -J-  to  those  currents  which  move 
in^  this  direction,  and  the  sign  —  to  those  moving  in  the 
contrary  direction.  As  to  the  electromotive  forces,  give 
them  -f-  or  —  signs,  according  as  they  cause  an  increase  or 
diminution  of  potential  in  the  given  direction. 


LAWS   OF   THE  ELECTRIC  CURRENT.  165 

The  application  of  Kirchhoffs  laws  to  a  combination  of  n 
conductors  gives  n  distinct  equations  between  the  currents, 
the  resistances,  and  the  electromotive  forces,  whence  we  can 
deduce  n  of  these  quantities  if  the  others  are  known. 

This  method  will  enable  us^  for  example,  to  determine 
the  currents  and  their  signs :  we  begin  by  supposing  arbi- 
trary directions  of  current — the  true  directions  will  be  given 
by  the  calculation  ;  a  positive  value  will  show  that  the  sup- 
posed direction  was  correct,  a  negative  value  will  indicate 
that  the  current  was  in  the  opposite  direction. 

122.  Application  to  Derived  Circuits.— The  combina- 
tion of  circuits  shown  in  Fig.  58  is  made  up  of  homogeneous 
conductors  having  resistances  r1,r.lJraJ  ending  in  the  points 
a  and  b,  which  are  connected  by  a  conductor  of  the  same 
nature,  including  a  chemical  source  of  electromotive  force 
/.  Let  r  be  the  resistance  of  this  part  of  the  circuit. 


FIG.  58. 

The  electromotive  force  produces  a  total  current  7,  which 
divides  itself  among  the  three  derived  branches  rl9  r9,  r% 
into  three  partial  currents  il  ,  /,  ,  i%  ,  such  that 


KirchhofTs  second  law  gives  the  equations 


1 66  ELECTRICIT  Y. 

Eliminating  successively  il ,  z, ,  z,  from  these  four  equa- 
tions, we  get 

e 


The  expression 


represents  the  combined  resistance  of  the  three  derived  con- 


ductors r1  ,  rv  and  r3 


In  general,  the  reciprocal  of  the  combined  resistance  of  a 
number  of  derived  conductors  is  equal  to  the  sum  of  the  recipro- 
cals of  the  resistances  of  the  component  conductors. 

123.  Wheatstone's    Bridge  or  Parallelogram. — The 

arrangement  in  Fig.  59  has  been  designed  by  Wheatstone  for 
the  purpose  of  measuring  electrical  resistances. 


FIG.  59. 

Suppose  there  are  six  conductors  having  resistances  a,  b, 
f,  d,  g,  p.  The  branch  r  contains  an  electromotive  force  e  ; 
the  branch  g  an  apparatus  to  indicate  the  passage  of  a  cur- 
rent. 

The  total  current  /  is  divided  into  partial  currents  which 
we  will  denote  by  the  capitals  A,  B,  F,  D,  G. 


ENERGY   OF   THE  ELECTRIC   CURRENT.  1  67 

The  application  of  Kirchhoff's  laws  gives  the  six  follow- 
ing equations: 

I-A-F  =o; 

A-G^-B  =o; 

E+G-&D  =o; 

aA 


Eliminating  A,  B,  F,  D,  we  have 

I(ad-bf) 


In  order  that  the  current  G  may  be  zero,  it  is  sufficient 
to  have. 

ad=bf    or     |=-£ 


ENERGY   OF  THE  ELECTRIC   CURRENT. 

124.  General  Expression.  —  In  consequence  of  the  defi- 
nition of  electric  potential,  when  a  quantity  q  of  electricity 
passes  from  a  potential  Ul  to  a  lower  potential  £7a  ,  the  work 
accomplished  is 


In  the  case  where  a  current  7  circulates  between  the  given 
points  the  work  per  second,  that  is,  the  electric  power  de- 
veloped by  the  current,  is  therefore 

(U,  -  U,)f. 

If  Ul  —  U^  represents  an  electromotive  force  E,  thermic 
or  chemical,  the  power  will  be  expressed  by 

EL 


1  68  ELECTRICIT  Y. 

125.  Application    to    the    Case   of   a   Homogeneous 
Conductor.     Joule's  Effect.  —  If  we  take  a   homogeneous 
conductor  of  resistance  R  traversed  by  a  constant  current 
/,  we  have 

(vl-uj/=m.  (§115.) 

The  work  developed  in  a  time  /  is 
W=I*Rt. 

Joule  has  experimentally  proved  that  this  work  is  en- 
tirely transformed  into  heat  in  the  conductor. 

One  of  the  most  beautiful  illustrations  of  Joule's  effect 
is  the  incandescent  electric  lamp,  in  which  the  current  heats 
a  carbon  filament  placed  in  a  glass  bulb  from  which  the  air 
has  been  exhausted  in  order  to  prevent  combustion. 

126.  Case    of    Heterogeneous    Conductors.      Peltier 
Effect.  —  Suppose    a    number    of    conductors   Rl  ,    R^  ,    ^3 
(§  120),  without  chemical  action  on  each  other;  denote  by  / 
the  current  which  flows  through  them,  by  El  and  E^  the  elec- 
tromotive  forces  of  contact.     By  Joule's   law  the  heat  de- 
veloped per  second  in  each  of  the  conductors  is  respectively 


At  the  points  of  junction  there  are,  in  addition,  abrupt 
variations  of  potential  El  ,  E  9  which  correspond  to  amounts 
of  electric  energy  .£,/,  EJ.  The  increase  of  energy  of  the 
current  will  be  negative  if  the  potentials  fall  in  the  direc- 
tion of  the  current  ;  it  will  be  positive  in  the  contrary  case. 

Peltier  has  shown,  in  the  first  case,  that  the  junction  is 
heated  ;  in  the  second,  that  it  is  cooled.  These  calorific 
variations  are  equal  and  contrary  to  the  variations  in  the 
energy  of  the  electric  flux.  This  phenomenon,  known  under 
the  name  of  Peltier  effect,  enables  us  to  measure  exactly  the 
electromotive  force  of  contact.  Contrary  to  the  Joule  effect, 


ENERGY   OF   THE  ELECTRIC  CURRENT.  169 

we  see  that  the  Peltier  effect  depends  on  the  direction  of 
the  current  and  that  it  changes  sign  with  the  current. 

To  show  the  Peltier  effect  it  is  necessary  to  take  steps  to 
prevent  the  heat  developed, in  the  conductors  by  the  Joule 
effect  from  hiding  the  variations  of  temperature,  generally 
feeble  at  the  points  of  junction.  This  is  effected  by  using 
feeble  currents  and  coating  the  junctions  with  some  readily 
fusible  substance,  such  as  wax.  The  wax  is  then  observed 
to  melt  when  a  current  flows  in  one  direction,  and  to  solidify 
when  it  flows  in  the  opposite  direction. 

From  the  law  of  successive  contacts  (§  112)  it  follows 
that  in  a  closed  circuit  where  no  differences  of  temperature 
are  maintained  by  external  sources  of  heat  the  algebraic 
sum  of  the  electromotive  forces  of  contact  is  zero,  and  con- 
sequently the  sum  of  the  Peltier  effects  is  also  zero. 

127.  Chemical  Effect  of  the  Current.  Faraday's  and 
BecquereTs  Laws. — When  an  electric  current  passes 
through  a.  compound  liquid,  by  means  of  conductors  or  elec- 
trodes dipping  into  the  liquid  and  kept  at  different  poten- 
tials, besides  the  heating  due  to  the  Joule  and  Peltier  effects, 
decomposition  of  the  liquid  is  observed  to  take  place. 

The  separated  elements  go  to  the  electrodes,  with  which 
in  certain  cases  they  enter  into  combination. 

This  decomposition  is  called  electrolysis,  and  the  decom- 
posed body  the  electrolyte.  The  electrode  having  the  highest 
potential,  by  which  the  current  enters,  is  the  positive  elec- 
trode or  anode;  the  other  is  the  negative  electrode  or  cathode. 
The  products  of  decomposition  are  the  ions. 

Electrolysis  takes  place  according  to  the  following  (Fara- 
day's) laws : 

I.  The  weights  of  the  ions  deposited  and  of  the  decomposed 
electrolyte  are  proportional  to  the  quantities  of  electricity  which 
have  passed  through  the  liquid. 


ELECTRICIT  Y. 

II.  When  a  number  of  electrolytes  are  traversed  by  the 
same  current,  the  weights  of  the  different  ions  set  free  are  to 
each  other  as  the  chemical  equivalents  of  these  ions. 

The  electrochemical  equivalent  of  an  ion  or  an  electrolyte 
is  the  weight  of  this  body  deposited  or  decomposed  per  unit 
quantity  of  electricity. 

BecquerePs  Law. — In  the  case  where  two  bodies  form 
various  combinations  with  each  other  the  decomposition  of 
these  different  combinations  is  dependent  on  the  negative 
element.  Thus  in  the  electrolysis  of  the  combinations 
/Wt ,  PN% ,  /yV8 ,  where  P  is  a  metal  and  N  a  metalloid,  unit 
quantity  of  electricity  sets  free  one  electrochemical  equiva- 
lent of  N  and  weights  of  P  equal  to  its  electrochemical 
equivalent  multiplied  by  I,  -J,  f. 

128.  Grothiiss'  Hypothesis. — The  fact  that  the  decom- 
position of  an  electrolyte  is  a  necessity  for  the  passage  of 
the  current  has  suggested  the  idea  that  the  ions  play  the 
same  part  as  the  pith-ball  in  convective  discharge. 

If  we  admit  that  the  molecules  of  the  electrolyte  are 
formed  of  groups  of  elements  having  opposite  charges  of 
electricity  (possibly,  according  to  Maxwell,  due  to  the  electro- 
motive force  of  contact),  then,  at  the  moment  of  introducing 
electrodes,  the  positive  elements  or  ions  will  turn  towards 
the  cathode  and  the  negative  elements  towards  the  anode. 
This  polarization  will  take  place  along  the  lines  of  force  of 
the  field  produced  in  the  liquid  by  the  electrodes.  If  the 
intensity  of  the  field  is  sufficient  to  overcome  the  chemical 
affinity  of  the  compound,  the  ions  near  the  electrodes  are  set 
free,  while  in  the  intermediate  molecules  of  the  liquid  there 
is  simply  an  exchange  of  elements.  We  have  thus  an  explana- 
tion of  the  reason  why  the  products  of  decomposition  only 
make  their  appearance  at  the  points  where  the  current  en- 
ters and  leaves  the  liquid. 


ENERGY  OF   THE  ELECTRIC  CURRENT.  I? I 

The  electric  charge  carried  per  second  by  the  positive 
ions  to  the  cathode  represents  the  amount  of  the  current, 
which  accounts  for  Faraday's  first  law.  To  account  for  the 
second  law  we  need  only  suppose  that  the  electronegative 
elements  of  different  electrolytes  have  all  the  same  electric 
charge. 

According  to  Clausius'  kinetic  theory,  the  molecules  are 
in  motion  and  their  impact  causes  their  disassociation  into 
the  component  atoms.  But  these  atoms  recombine  with 
those  liberated  from  the  adjoining  molecules,  so  that  there 
are  continual  changes  in  all  directions.  The  electric  current 
causes  an  orientation  of  these  movements  and  brings  about 
a  final  decomposition  at  the  electrodes. 

129.  Application  of  the   Conservation  of  Energy  to 

Electrolysis.  Voltaic  Cell. — The  phenomenon  of  electroly- 
sis can  be  considered  from  the  point  of  view  of  the  con- 
servation of  energy. 

In  an  endothermic  electrolytic  reaction  which  absorbs 
energy,  as  is  the  case  when  acidulated  water  is  decomposed 
between  platinum  electrodes,  the  effective  energy  of  the 
current  is  diminished  ;  a  fall  of  potential  takes  place  in  the 
direction  of  the  current,  equal  to  what  is  called  the  elec- 
tromotive force  of  polarization  of  the  electrolyte.  This 
electromotive  force  is  negative  (§  119),  and  tends  to  produce 
a  counter-current.  The  existence  of  this  electromotive  force 
can  be  shown  by  connecting  the  platinum  conductors,  imme- 
diately after  electrolysis,  to  an  apparatus  for  showing  the 
passage  of  a  current  and  then  closing  the  circuit.  A  current 
will  be  observed  directed  from  the  cathode  to  the  anode  in 
the  electrolyte,  and  at  the  same  time  the  liberated  elements, 
oxygen  and  hydrogen,  will  recombine. 

Lord  Kelvin  has  shown  that  the  electromotive  force  of 
polarization  can  be  calculated  when  the  energy  of  combina- 


ELECTRICITY. 

tion  of  the  electrolyte  is  known.  In  fact,  if  no  secondary 
action  takes  place,  the  electrical  energy  absorbed  (represented 
by  the  product  ie  of  the  electromotive  force  of  polarization 
into  the  current)  is  equal  to  the  heat  of  combination  of  the 
weight  of  electrolyte  decomposed  per  second,  expressed  in 
absolute  units.  Let  z  be  the  electrochemical  equivalent  of 
the  electrolyte,  h  the  heat  of  combination  of  unit  weight  of 
the  same.  We  then  have 

ie  =  zhi  ; 
whence 


This  expression  gives  the  minimum  difference  of  potential 
between  the  electrodes  necessary  to  produce  decomposition. 

From  these  considerations  we  deduce  a  means  of  separating 
the  elements  of  several  electrolytes  mixed  together.  Sup- 
pose, for  example,  a  solution  containing  sulphate  of  zinc  and 
sulphate  of  copper  ;  as  the  heat  of  combination  of  the  sec- 
ond salt  is  less  than  that  of  the  first,  we  can,  by  suitably 
graduating  the  potential  difference  of  the  electrodes,  deposit 
first  copper  and  then  zinc  upon  the  cathode. 

There  are  cases  where  the  ions  react  on  the  electrodes, 
giving  rise  to  new  compounds.  The  energy  set  free  in  these 
reactions  must  be  taken  into  account  in  calculating  the  elec- 
tromotive force  necessary  for  decomposition. 

Take  the  example  of  the  electrolysis  of  a  solution  of  cop- 
per sulphate  between  copper  electrodes.  The  liberated  cop- 
per will  go  to  the  cathode  and  the  acid  to  the  anode,  which 
it  will  dissolve  equivalent  for  equivalent.  The  reaction  at 
the  electrodes  thus  neutralizes  the  chemical  effect  of  the 
current,  so  that  the  electromotive  force  of  decomposition  is 
zero.  The  whole  energy  of  the  current  is  used  in  the  Joule 
effect,  that  is,  in  heating  the  bath. 


ENERGY  OF   THE  ELECTRIC  CURRENT.  1/3 

Suppose  that  water  acidulated  with  sulphuric  acid  is  de- 
composed between  a  zrnc  anode  and  a  copper  cathode.  The 
hydrogen  will  be  deposited  on  the  copper,  while  the  oxygen 
will  form  oxide  of  zinc  with,  the  anode,  which  will  then  be 
dissolved  in  the  state  of  sulphate  of  zinc.  But  as  the  heat 
of  combination  of  zinc  sulphate  is  higher  than  that  of  sul- 
phuric acid,  a  quantity  of  energy  is  consequently  liberated 
which  shows  itself  by  an  increase  of  potential  in  the  direction 
of  the  current  equal  to  the  effective  difference  of  the  elec- 
tromotive forces. 

This  electromotive  force  is  E  =  ph  —  p'h'  ,  ph  denoting  the 
heat  of  formation  of  one  electrochemical  equivalent  of  zinc 
sulphate,  p'h'  that  of  one  equivalent  of  sulphuric  acid. 

Such  a  combination,  called  voltaic  element  or  couple,  is  a 
source  of  electricity  ;  and  if  we  connect  the  two  electrodes  by 
a  copper  wire  it  will  be  found  that  it  is  traversed  by  a  cur- 
rent flowing  from  the  zinc  to  the  copper  in  the  electrolyte, 
and  from  the  copper  to  the  zinc  in  the  external  circuit  formed 
by  the  wire. 

The  direction  of  the  current  in  the  external  circuit  shows 
that  the  copper  is  at  a  higher  potential  than  the  zinc,  whence 
the  name  of  positive  pole  or  plate  given  to  the  plate  of  cop- 
per. The  zinc  plate  is,  conversely,  called  the  negative  pole 
or  plate. 

The  current  is 


where  E  represents  the  electromotive  force  and  R  the  re- 
sistance of  the  circuit,  including  the  resistance  of  the  liquid 
as  well  as  that  of  the  wire  and  the  electrodes. 


174  ELECTRICITY. 


THERMO-ELECTRIC   COUPLES. 

130.  Seebeck  and  Peltier  Effects. — Seebeck  has  proved 
(§113)  the  production  of  an  E.  M.  F.  in  a  chain  of  metals 
whose  junctions  are  maintained  at  unequal  temperatures. 
Thus  on  forming  a  circuit  of  an  iron  and  a  copper  wire  the 
latter  including  a  galvanometer  coil,  and  on  raising  the  tem- 
perature of  one  of  the  points  of  junction,  a  current  is  set  up 
in  the  system  going  from  the  copper  to  the  iron  across  the 
hot  junction.  Such  a  circuit  is  called  a  thermo-electric 
couple. 

The  Seebeck  effect  is  reversible,  as  has  been  shown  by 
Peltier ;  when  a  current  due  to  an  outside  E.  M.  F.,  traverses 
the  junction  of  the  two  metals,  from  the  copper  to  the  iron, 
the  junction  is  cooled  ;  if  the  direction  of  the  current  is  re- 
versed, it  grows  hot.  This  phenomenon  is  distinct  from  the 
Joule  effect ;  but,  as  the  two  effects  occur  simultaneously, 
certain  precautions  must  be  taken  in  order  to  distinguish  one 
from  the  other. 

The  development  of  heat  due  to  the  Peltier  effect  is  pro- 
portional to  the  first  power  of  the  current,  while  the  Joule 
effect  depends  on  its  square  ;  it  is  therefore  advantageous  to 
employ  feeble  currents  in  order  to  distinguish  the  two 
actions. 

According  to  Maxwell,  the  Peltier  effect  is  a  measure  of 
the  E.  M.  F.  of  contact,  §  107.  In  fact  the  heat  developed  at 
the  junction  by  a  current  i  in  one  second  is  expressed  by  the 
product  ei,  in  which  e  represents  the  difference  of  potential 
which  is  set  up  on  the  contact  of  the  given  bodies.  If  this 
heat  n  is  given  in  gramme-degrees,  e  in  volts,  and  i  in  am- 
peres, we  have  4.2^  =  ie  ;  whence 

4.2«       . 

€  =  2—r-   VOltS. 


THERMO-ELECTRIC  COUPLES.  1/5 

The  values  found  by  this  means  depend  on  the  absolute 
temperature  of  the  junction  e  =  f  (T) ;  they  are  very  slight 
compared  with  the  potential  differences  observed  on  con- 
necting to  the  terminals  of  an*electrometer  two  points  of 
the  conductors  taken  on  each  sig(e  of  the  junction  and  close 
to  it. 

Thus  the  E.  M.  F.  of  contact  of  zinc  and  copper  measured 
by  the  first  method  is,  at  25°  C,  0.00045  v0^  while  the 
electrometer  indicates  about  0.8  volt  ;  but  Maxwell  has 
observed  that,  in  the  latter  case,  we  have  not  only  to  do 
with  the  contact  Zn  \  Cu,  but  that  these  metals  form,  with 
the  air  which  separates  the  fixed  parts  from  the  movable 
parts  of  the  electrometer,  a  closed  chain  : 

Zn  |  Cu  +  Cu  |  air  -f-  air    |  Zn. 

Now  the  E.  M.  F.  of  contact  of  the  air  with  the  metallic 
parts  of  the  instrument,  by  virtue  of  which  they  assume 
electric  charges,  may  be  much  greater  than  that  which  is 
developed  in  an  entirely  metallic  contact,  which  would  ex- 
plain the  observed  anomaly. 

In  the  thermo-electric  series  a  metal  is  said  to  be  positive 
with  regard  to  another  when  the  E.  M.  F.  of  contact  is 
directed  from  the  first  to  the  second  across  the  heated  junc- 
tion. 

The  E.  M.  F.  which  is  set  up  in  a  metallic  arc  formed  of  two 
dissimilar  metals  depends,  as  might  be  expected,  on  the  heat 
communicated  to  one  of  the  junctions,  and  the  simplest  way 
is  to  express  this  E.  M.  F.  as  a  function  of  the  difference  be- 
tween the  temperatures  0,  0'  of  the  two  junctions  of  the 
metals  forming  the  circuit.  But  we  have  seen  that  this 
E.  M.  F.  is  also  dependent  on  the  absolute  temperature  of  the 
two  junctions,  or,  in  other  words,  on  the  mean  of  their  tem- 
peratures : 


1 76  ELECTRICIT  Y. 

Thus,  when,  in  the  case  of  the  above-mentioned  couple, 
we  make  one  of  the  junctions  continually  hotter  while 
keeping  the  other  at  a  constant  temperature,  the  thermo- 
electric current  increases  to  a  maximum,  then  decreases, 
becomes  zero,  and  ends  by  changing  direction. 

131.  Kelvin  Effect. — In  seeking  for  the  cause  of  this 
peculiarity  Lord  Kelvin  discovered  that  the  E.  M.  F.  which 
gives  rise  to  the  current  is  not  situated  at  the  junctions 
alone,  as  was  long  believed,  but  that  the  homogeneous  wires, 
unequally  heated,  which  form  the  circuit  are  also  the  seat 
of  electromotive  forces. 

Thus  in  a  metal  bar  an  unequal  distribution  of  tempera- 
ture causes  differences  of  potential  between  the  various 
points.  If  the  temperature  rises  from  one  end  of  the  bar  to 
the  other,  a  continual  rise  of  potential  is  observed  for  some 
metals,  while  others  give  a  fall  of  potential  in  the  direction 
corresponding  to  the  rise  of  temperature. 

According  to  M.  Leroux,  lead  is  the  only  metal  in  which 
similar  electric  phenomena  are  not  manifested  when  various 
parts  of  a  piece  of  this  substance  are  put  in  different  thermal 
conditions. 

These  E.  M.  F.'s  combine  with  those  which  are  set  up  at 
the  points  of  junction  and  give  a  resultant  E.  M.  F.,  whose 
ratio  to  the  resistance  of  the  circuit  is  the  strength  of  the 
thermo-electric  current.  If  the  sum  of  the  potential  differ- 
ences set  up  in  the  metals  is  opposite  to  the  sum  of  the 
potential  differences  at  the  junctions,  it  may  be  assumed  that 
there  exist  temperatures  for  which  these  sums  are  equal,  and 
the  thermo-electric  current  ceases.  The  mean  of  the  tem- 
peratures of  the  junctions  for  which  this  phenomenon 
occurs  is  called  the  neutral  temperature  or  temperature  of 
reversal. 

The  properties  discovered  by  Lord  Kelvin,  and  which  are 


THERMO-ELECTRIC   COUPLES.  1 77 

called  the  Kelvin  effect,  are,  from  a  certain  point  of  view, 
reversible. 

Thus,  when  a  current  is  passed  through  a  wire  whose  ends 
are  maintained  at  different  tqmPperatures,  and  which  conse- 
quently has  a  distribution  of  potential  of  its  own,  the  current 
cools  the  wire  if  it  is  directed  towards  increasing  potentials, 
and  heats  it  in  the  contrary  case. 

In  order  to  exhibit  this  effect  independently  of  the  Joule 
effect  proceed  as  follows :  A  metal  bar  is  heated  towards 
the  middle,  while  its  extremities  are  kept  at  o°  in  melting 
ice.  When  a  current  passes  in  the  bar,  it  is  found  that  points 
situated  symmetrically  with  respect  to  the  middle  are  not  at 
the  same  temperature  ;  in  fact  there  is  a  greater  heating  in 
one  of  the  halves  where  the  Joule  and  Kelvin  effects  are 
added  together  ;  in  the  other  half  the  Kelvin  effect  absorbs 
part  of  the  heat  due  to  the  Joule  effect.  The  action  is  the 
same  as  if  there  were  a  transference  of  heat  in  the  bar ;  the 
transfer  is  made  in  the  direction  of  the  current  for  certain 
metals,  and  in  the  opposite  direction  for  others.  Lead  is 
the  only  metal  which  preserves  a  perfect  symmetry  as  re- 
gards the  distribution  of  the  temperature. 

It  is  interesting  to  observe  that,  in  a  thermo-electric  chain 
whose  junctions  are  maintained  at  unequal  temperatures, 
the  algebraic  sum  of  the  potential  differences  is  zero,  as  in- 
deed is  the  case  in  every  closed  electric  circuit.  It  follows 
that  the  heating  observed  at  the  points  where  the  current 
experiences  a  drop  in  potential  exactly  compensates  the 
corresponding  cooling  at  the  rises  of  potential;  in  other 
words,  the  total  heat  produced  in  the  circuit  by  the  Pel- 
tier and  Kelvin  effects  is  zero.  This  is  not  the  case  with 
the  heat  corresponding  to  the  Joule  effect,  whose  value, 
necessarily  positive,  represents  the  quantity  of  heat  supplied 
at  the  hot  jnnction,  excepting  for  losses  by  radiation  and 
conductivity. 


1 78  ELECTRICIT  Y. 

132.  Laws  of  Thermo-electric  Action. — The  two  fol- 
lowing laws  were  discovered  experimentally  by  Becquerel : 

Law  of  Successive  Temperatures. — In  a  thermo-electric 
couple  formed  of  two  dissimilar  bodies  the  E.  M.  F.  corre- 
sponding to  two  temperatures  #,  and  0,  of  the  junctions  is 
equal  to  the  algebraic  sum  of  the  E.  M.  F.'s  corresponding 
to  the  temperatures  0,  and  #  on  the  one  hand,  and  0  and  #, 
on  the  other. 

Law  of  Intermediary  Metals. — If  two  metals  in  a  circuit 
are  separated  by  one  or  more  intermediary  metals,  all  kept 
at  the  same  temperature,  the  E.  M.  F.  is  the  same  as  if 
the  two  metals  were  directly  united  and  their  junction  raised 
to  the  same  temperature  ;  consequently  the  solder  placed 
between  two  metals  is  without  effect  on  the  E.  M.  F.  of  the 
couple. 

These  laws  are  supplemented  by  those  of  Kelvin  and 
Tait,  which  may  be  announced  as  follows  : 

Kelvin  s  Law. — If  the  extremities  of  a  homogeneous  bar 
are  kept  at  temperatures  6  and  8' ',  an  E.  M.  F.  exists  in  the 
bar  proportional  to  0'  —  8,  the  coefficient  of  proportionality, 
itself  variable  with  the  temperature,  is  called  by  Kelvin  the 
specific  heat  of  electricity. 

These  laws  being  granted,  let  us  consider  a  couple  formed 
of  the  metals  A  and  B,  whose  junctions  are  maintained  at 
temperatures  B  and  Q'\  let  U  and  V  be  the  sudden  variation 
of  potential  at  junctions,  cr  and  a*'  the  specific  heats  of  elec- 
tricity of  A  and  B.  The  total  E.  M.  F.  will  be 

e=U-U' 

Taifs  Law. — The  specific  heat  of  electricity  of  a  body  is 
proportional  to  its  absolute  temperature,  cr  =  K6. 

133.  Thermo-electric   Powers. — The  E.  M.  F.'s  which 
are  set  up  in  thermo-electric  couples  by  the  effect  of  progres- 


THERMO-ELECTRIC   COUPLES.  1 79 

sive  variations  of  temperature  are  determined  by  observing 
the  currents  which  result  from  them  in  circuits  of  known  re- 
sistance. The  increase  of  resistance  caused  by  the  rise  of 
temperature  of  one  of  the  junctions  is  rendered  negligible 
by  introducing  into  the  circuit  ait  large  supplementary  resist- 
ance, which  may  be  of  the  same  substance  as  one  of  the 
metals  in  the  couple,  or  of  a  different  substance  on  condition 
that  it  is  kept  at  the  same  temperature  in  all  its  points  (law 
of  intermediary  metals). 

Suppose  that  one  of  the  junctions  be  kept  at  an  invariable 
temperature  by  immersion  in  melting  ice,  for  example,  and 
the  other  junction  raised  to  increasing  temperatures  by  im- 
mersion in  a  heated  bath  containing  a  thermometer.  The 
difference  of  temperature  and  the  total  E.  M.  F.'s  are  to  be 
observed  simultaneously.  To  represent  the  phenomenon 
graphically  the  values  of  the  former  may  be  made  abscissae 
and  those  of  the  latter  ordinates ;  the  curves  thus  drawn  are 
very  sensibly  parabolas  with  vertical  axes,  their  apex  corre- 
sponding to  the  temperature  of  reversal  (Gaugain). 

Their  equation  is  of  the  form 


From  this  we  deduce 
d 


This  derivative,  which  is  the  angular  coefficient  of  the  tan- 
gent to  the  parabola,  is  the  E.  M.  F.  corresponding  to  a 
difference  of  temperature  of  i°  between  the  junctions  at  the 
mean  temperature  0.  This  value  has  by  Lord  Kelvin  been 
given  the  name  of  thermo-electric  power  of  the  couple  at  the 
given  temperature. 


1  8O  ELECTRICIT  Y. 

At  the  neutral  temperature  the  tangent  is  parallel  to  the 
axis  of  the  abscissae,  whence 


The  total  E.  M.  F.  corresponding  to  temperatures  0  and  6' 
at  the  junctions  is  given  by  the  knowledge  of  the  coefficients 
a  and  b : 


This  formula  shows  that,  when  the  temperatures  B  and  0' 
are  equidistant  from  /„  ,  the  E.  M.  F.  is  zero. 

In  order  to  determine  the  pairs  of  parameters  a  and  b  or 
b  and  tn  we  need  only  perform  experiments  at  temperatures 
/  and  t'  ,  ^  and  //  ;  we  thus  get  two  equations  in  which  the 
quantities  sought  are  the  only  unknown  ones: 


In  order  to  represent  graphically  the  variations  of  the 
thermo-electric  power  of  a  coupled  \  B  with  the  temperature, 
we  need  only  draw  the  right  line  MM  ',  whose  equation  is 


For  another  couple  A  \  C  we  shall  have  a  second  right  line 
NN'  usually  cutting  the  first. 


THERMO-ELECTRIC  COUPLES. 


181 


Now  by  the  law  of  intermediary  metals  the  thermo-elec- 
tric power  of  the  couple  C  \  B  will  be  given  by  the  difference 
of  the  ordinates  of  MM'  and  NNr. 


FIG.  60. 

Knowing  the  parameters  of  the  two  couples  A  \  B  and 
A  |  C, 


(gj  =***% 

\atrjA    c 


we  deduce  those  of  the  couple  C  \  B  from  the  formula 


We  need  therefore  only  draw  diagrams  of  the  thermo- 
electric powers  of  all  the  metals  taken  separately  with  one 
of  their  number  in  order  to  learn  the  values  of  the  thermo- 
electric powers  of  all  the  metals  taken  in  pairs  in  any  com- 
bination. 

Lead  is  generally  adopted  as  the  metal  of  comparison,  be- 
cause its  specific  heat  of  electricity  is  zero. 

In  these  diagrams  the  intersection  of  two  right  lines  has 
as  its  abscissa  the  value  of  the  temperature  of  reversal. 

It  will  be  observed  that  the  E.  M.  F.  of  a  couple  A  |  B 
between  two  temperatures  6  and  &  is  expressed  by 


1 82  ELECTRICITY. 

being  represented  by  the  area  included  between  the  right 
line  MM' y  the  axis  of  abscissas  and  the  extreme  ordinates 
corresponding  to  6  and  0'.  Likewise  the  E.  M.  F.  of  C  \  B, 
between  the  same  temperatures,  is  represented  by  the  area 
MM'N'N.  This  area  can  also  represent  the  work  done  by  a 
quantity  of  electricity  equal  to  one  coulomb  traversing  the 
circuit  C  |  B. 

We  give  here  the  values  of  the  parameters  a  and  by  which 
enable  the  thermo-electric  powers  of  various  bodies,  taken 
with  lead,  to  be  calculated  in  microvolts : 

a  b 

Copper —    1.34  —0.0094 

Alloy  (90  Pt  +  10  Ir)..     —    5.90  +  i -Oi 33 

Iron   —17.15  +0.0482 

German-silver -["11.94  +0.0506 

From  these  figures  it  is  seen  that  an  iron-german-silver 
couple  has  a  thermo-electric  power  of 

(—  29.09  —  0.00246)  microvolts. 

The  current  goes  from  the  german-silver  to  the  iron  across 
the  hot  junction.  The  E.  M.  F.  for  the  temperatures  o°  and 
200°  at  the  junctions  is  5.866  millivolts. 

These  results  show  that  thermo-electric  couples  only  pro- 
duce extremely  feeble  E.  M.  F's,  and  that  it  is  consequently 
necessary  to  join  up  a  large  number  of  them  in  series  in 
order  to  obtain  differences  of  potential  comparable  to  those 
obtained  in  hydro-electric  batteries.  It  is  true  that,  as  the 
couples  are  formed  of  very  good  conductors,  they  are  ca- 
pable of  producing  tolerably  strong  currents  in  an  external 
circuit  of  small  resistance. 

Various  bodies  give  E.  M.  F.'s  very  much  higher  than 
those  of  the  common  metals,  but  they  cannot  stand  as  high 
temperatures.  According  to  Becquerel,  at  a  temperature  of 


THERMO-ELECTRIC  COUPLES.  183 

50°  C.  the  thermo-electric  power  of  the  bismuth-lead  couple 
is  -|-  40  microvolts  and  that  of  the  fused  copper-sulphate- 
lead  couple  is  —  352  microvolts.  Antimony-zinc  alloy  (in 
equal  proportions)  gives  witn  lead  —  98  microvolts.  The 
conductivity  of  these  different  Bodies  is  very  inferior  to  that 
of  the  metals,  and  it  is  necessary  to  use  them  in  the  form  of 
tolerably  thick  bars. 

134.  Thermo-electric  Pile. — To  form  a  thermo-electric 
pile  a  chain  is  made  in  which  the  alternate  links  are  formed 
of  one  of  the  two  metals  chosen  to  form  a  couple,  and  all 
the  odd  (or  all  the  even)  junctions  are  heated. 

In  order  not  to  need  to  use  as  many  heaters  as  there  are 
couples  the  chain  is  folded  zigzag,  the  consecutive  links  be- 
ing isolated  with  asbestos;  in  this  way  a  solid  block  is  ob- 
tained, with  the  even  junctions  on  one  side  and  the  odd  ones 
on  the  opposite  side  ;  then  only  one  source  of  heat  is  needed 
to  heat  all  the  junctions  on  one  side.  The  opposite  junctions 
can  be  cooled  by  a  current  of  air  ;  they  are  also  often  fur- 
nished with  expansions  intended  to  aid  the  radiation  of  the 
heat  transmitted  across  the  couples  by  conductivity  ;  for 
this  purpose  are  used  thin  sheets  of  copper  or  iron,  blackened 
in  order  to  increase  their  emissive  power. 

It  frequently  happens  that  the  substances  used  to  make 
the  couples  are  not  very  capable  of  supporting  the  direct 
action  of  the  flames  which  heat  the  junctions.  In  this  case 
the  latter  are  covered  with  a  solid  envelope  upon  which  the 
flame  plays,  and  which  transmits  the  heat  to  the  couples  by 
conductivity.  This  arrangement  has  also  the  advantage  of 
rendering  the  variations  of  temperature  in  the  couples  less 
sudden  when  the  fire  is  lighted  or  extinguished,  and  con- 
sequently of  diminishing  the  disintegration  which  takes 
place,  in  consequence  of  these  sudden  changes,  in  the  bars 
of  alloy  employed. 


ELECTROMAGNETISM. 

MAGNETIC  PHENOMENA  DUE  TO  CURRENTS. 

J35-  Oersted's  Discovery.  —  Oersted  established  in 
1820  the  action  of  an  electric  current  upon  a  magnet- 
needle.  This  discovery  was  the  point  of  departure  for  the 
theory  of  electromagnetism,  which  was  established  almost 
entirely  by  Ampere.  According  to  the  practical  rule 
pointed  out  by  this  physicist,  the  north  pole  of  the  needle 


FIG.  61. 

tends  to  move  towards  the  left  hand  of  an  observer  who 
looks  at  the  needle  when  he  is  placed  in  the  direction  of 
the  current,  so  that  it  enters  at  his  feet  (Fig.  61).  This 
fundamental  action  shows  that  the  current  produces  a  mag- 
netic field,  which  fact  can  be  also  shown  by  the  use  of  iron- 
filings  (§  47). 

Upon  dusting  iron-filings  on  a  sheet  of  paper  traversed 
by  a  current  perpendicular  to  the  plane  of  the  paper  it  is 
seen  that  the  particles  form  circles  whose  centre  is  in  the 
axis  of  the  conductor.  A  magnetic  pole  free  to  move 

184 


MAGNETIC  PHENOMENA    DUE    TO    CURRENTS.      185 

about  the  conductor  would  consequently  tend  to  turn 
around  it.  The  direction  of  this  movement  can  be  deter- 
mined by  Ampere's  rule,  or  by  that  of  Maxwell,  which  is 
often  more  convenient  to  apply.  The  direction  of  rotation 


FIG.  62. 

of  the  north  pole  and  the  direction  of  the  current  are  indi- 
cated by  the  relative  movements  of  rotation  and  translation 
of  a  corkscrew. 

The  circular  form  of  the  magnetic  lines  of. force  due  to 
a  rectilinear  current  explains  why  a  magnet-needle  tends  to 
place  itself  transversely  to  the  current,  so  that  its  magnetic 
axis  may  be  tangent  to  the  line  of  force  which  passes 
through  its  support. 

136.  Magnetic  Field  due  to  an  Indefinite  Rectilinear 
Current. — The  intensity  at  different  points  of  the  field 
may  be  studied  by  the  method  of  oscillations  (§  47).  By 
applying  this  method,  Biot  and  Savart  have  found  that  the 
intensity  of  the  field  due  to  a  rectilinear  current,  sufficiently 
long  and  distant  from  the  rest  of  the  circuit  to  be  consid- 
ered as  indefinite,  is  proportional  to  the  current  and  in- 
versely proportional  to  the  distance  from  the  conductor. 
The  direction  of  the  field  is  normal  to  the  plane  passing 
through  the  conductor  and  the  point  under  consideration. 
The  force  exercised  on  a  positive  pole  m  can  therefore  be 

expressed  by 

kim 


1  86  ELECTROMA  GNE  TISM. 

The  intensity  of  the  field  at  a  distance  /  is  consequently 


As  the  reaction  is  equal  and  contrary  in  direction  to  the 
action,  a  pole  m  exercises  upon  the  current  a  force  equal  to 

kim 


This  force  is  directed  towards  the  right-hand  of  Am- 
pere's manikin  when  he  faces  the  pole,  or  towards  his  left 
if  he  is  looking  in  the  direction  of  the  lines  of  force  emerg- 
ing from  the  pole. 

137.  Laplace's  Law.  —  Biot  has  investigated  the  action 
of  a  current  traversing  two  indefinite  rectilinear  conductors, 
AB,  AC,  placed  at  an  angle,  upon  a  pole  m  situated  on  the 


FIG.  63. 


line  bisecting  the  angle  (Fig.  63).     He  found  that  the  force 
can  be  represented  by 


k  being  a  coefficient  of  proportion,  /  the  distance  from  the 
pole  P  to  the  apex  A,  a  the  half-angle  between  the  conduc- 


MAGNETIC  PHENOMENA    DUE    TO    CURRENTS.      1  8? 

tors.     This  expression  reduces  to  that  of  the  preceding  par- 
agraph when  ex  =  90°.   . 

The  direction  of  the  force  is  normal  to  the  plane  of  the 
two  conductors.  Laplace  has  Deduced  from  this  expression 
the  action  of  an  element  of  current  on  a  pole.* 

ll 

By  reason  of  symmetry,  the  effect  of  one  of  the  branches 
ABis 


(i) 


a  being  the  angle  made  by  AB  with  PA. 

Prolong  the  branch  BA  by  a  quantity  AA'  =  ds,  and 
find  the  action  of  this  element  of  current  on  the  pole  situ- 
ated at  P. 


FIG.  64. 

It  will  be  observed  that  F  is  a  function  of  two  variables, 
r  and  a,  which  determine  the  relative  positions  of  m  and  ds. 
We  can  then  write  the  identity 

,.,      dF  ,         (dFda  .   dFdr\  , 
dF=  —  ds  =    —  — +  -—  — - )  ds.     .     .     .  (2) 
ds  \dads        dr  dsJ 

In  order  to  obtain  the  expression  for  df,  we  need  only 
substitute  in  (2)  the  values  of  the  four  derivatives,  deducing 


*The  following  demonstration  is  due  to  M.  de  Weydlich,  formerly  assist- 
ant at  the  Liege  Electrotechnical  Institute. 


1  88  ELECTROMAGNETISM. 

them  from  the  results  of  experiments  and  from  geometrical 
considerations.  Drawing  from  the  point  P  as  centre  the 
arc  AA",  and  observing  that  the  angle  APA"  =  da,  since  it 
is  the  increase  of  the  angle  between  the  directions  of  AB 
and  PA,  we  will  have  in  the  infinitely  small  triangle  AA'  A" 

A  A"  =  ds  sin  a  =  Ida, 
whence 


da  _  sin  a 
d7=    ~~T 


and 


whence 

dl 

—  -  =  —  cos  a. 
ds 

On  the  other  hand,  equation  (i)  gives  directly 
dF         k'im       i 

ac**   /    ~^ 

cos3-. 

dF  k'im  t  a 
—  r-f  =  --  ^  —  tan  — 
dl  ?  2- 

Substituting  these  expressions  in  (2),  we  find 

i 

dF  =  —  TJ-  [j"-       -  sin  a  -\-  tan  -.  cos  on  d^ 
cos'f 


.  ,     .    u  ,  x 

=  --  sin  ^d^  =  --    dj  sm  (/,  ds). 


MAGNETIC  PHENOMENA    DUE    TO    CURRENTS.  '"  189 

The  elementary  force  is  normal  to  the  plane  of  the  cur- 
rent and  of  the  pole.  .  If  we  consider  the  reaction  of  the 
pole  on  the  element  ds,  we  find  it  directed  towards  the 
right  hand  of  Ampere's  manikin  when  placed  in  the  direc- 
tion of  the  current  and  facing  r<  the  pole  (§  136),  or  towards 
his  left  if  turned  so  as  fo  face  in  the  direction  of  the  lines  of 
force  produced  by  the  pole. 

138.  Action  of  a  Magnetic  Field  on  an  Element  of 
Current.  —  It  will  be  noticed  that  in  Laplace's  law  the  fac- 

7/2 

tor  -j-  represents  the  intensity  of  the  field  JC  due  to  the  pole 

^2 

m,  at  the  point  where  the  current-element  is  situated.     We 
can  therefore  write 

dF  =  ki  JC  ds  sin  (JC,  ds). 

It  is  easy  to  generalize  the  law  for  the  case  of  a  r  umber 
of  poles. 

The  total  force  dF  is  the  product  of  kids  by  the  resultant 
of  the  terms  such  as 

m     .     ,*   ,\ 

—  sin  (/,  ds), 


which  represents  the  product  of  the  intensity  of  the  field  JC 
by  the  sine  of  the  angle  between  the  direction  of  the  field 
and  the  direction  of  the  element,  since  the  projection  of  the 
resultant  is  equal  to  the  sum  of  the  projections  of  the 
components. 

Consequently  the  resultant  is,  in  absolute  value, 

dF=  ki  dsW,  sin  (JC,  ds). 

From  what  we  have  learned  in  the  preceding  paragraph, 
this  force,  applied  to  an  element  of  current,  is  normal  to  the 
plane  of  the  current  and  the  field,  and  directed  towards  the 


190 


ELECTROMA  ONE  TISM. 


left  of  Ampere's  manikin   when  he  faces  in  the  direction 
of  the  lines  of  force  of  the  field. 

Fleming  has  pointed  out  another  way  of  determining 
the  direction  of  the  electromagnetic  force.  If  the  thumb 
and  first  two  fingers  of  the  left  hand  be  pointed  in  three 
directions  perpendicular  to  each  other,  pointing  the  fore 
and  middle  fingers  respectively  in  the  direction  of  the 
magnetic  lines  of  force  and  the  current,  then  the  thumb 
points  in  the  direction  in  which  this  last  tends  to  be 
displaced. 

139.  Work  due  to  the  Displacement  of  an  Element 
of  Current  under  the  Action  of  a  Pole. — Suppose  a  cur- 
rent-element ds  =  ab  (Fig.  65)  acted  on  by  a  pole  situated 
in  a  point  P. 


FIG.  65. 

The  electromagnetic  force  tends  to  displace  the  element 
ds  perpendicularly  to  the  plane  /,  d5  following  a  direction 
af.  If  the  element  moves  in  a  direction  ag,  the  work  per- 
formed during  a  displacement  ds'  —  ag  is  equal  to  the  prod- 
uct of  the  force  by  the  projection  of  the  displacement  along 
the  direction  of  the  force. 


MAGNETIC  PHENOMENA    DUE    TO    CURRENTS. 

We  then  have 


dW=          dssm(Pal>)ds'  cos(gaf).       .     .     (i) 

t 

Now  construct  a  parallelogram  on  ag  and  ab  ;  pass 
through  af  a  plane  fam  normal  to  /and  determine  the  in- 
tersections e,  n  of  this  plane  with  the  right  lines  gP,  dP. 

As  ag  and  ab  are  infinitely  small  in  comparison  with  r, 
the  plane  gdP  is  normal  to  fam  and  contains  the  right  line 
gf  which  is  projected  in  ag  on  af. 

Equation  (i)  can  consequently  be  written 

~X^Xtf. 

But  the  product  am  X  af  measures  the  surface  of  the 
parallelogram  aenm,  of  which  af  is  the  altitude,  this  paral- 
lelogram being  capable  of  being  considered  as  the  projec- 
tion of  agdb  on  a  sphere  of  radius  /  and  centre  P. 

Dividing  this  projection  by  F,  we  obtain  the  projection 
on  a  sphere  of  unit  radius,  or  the  solid  angle  subtended  by 
the  parallelogram  agdb,  that  is  to  say,  the  area  described  by 
the  element  ds.  Calling  this  solid  angle  do?, 

d  W  —  kimd  GO. 

140.  Work  Due  to  the  Displacement  of  a  Circuit  under 
the  Action  of  a  Pole.  —  To  find  the  work  performed  by  a 
current  of  finite  length  displaced  under  the  action  of  a  pole, 
we  need  only  consider  the  sum  of  such  terms  as  kimdoo. 
We  find  the  product  of  kim  by  the  solid  angle  subtended  at 
the  pole  by  the  surface  described  by  the  given  current. 

In  the  case  of  a  closed  circuit,  abgd.  Fig.  66,  which  as- 
sumes a  position  a'b'g'd'  under  the  action  of  a  pole  situated 


1 92  ELECTROMA  GNE  TISM. 

in  the  direction  of  the  reader,  we  can  consider  separately 
the  segments  abg,  gda  traversed  by  opposite  currents.  The 
work  accomplished  by  abg  is  proportional  to  the  solid  angle 


subtended  by  the  area  aa'b'g'gb  ;  that  by  adg  is  propor- 
tional to  the  apparent  surface  of  aa'd'g'gd.  The  resultant 
work  will  be  proportional  to  the  difference  between  these 
apparent  surfaces,  that  is,  to  the'  difference  between  the 
solid  angles  subtended  by  the  two  contours  a'  b'  g'  d'  ,'  abgd  of 
the  circuit. 

It  follows  from  the  preceding,  that  to  bring  a  pole  m 
from  an  infinite  distance  to  a  point  at  which  the  outline  of 
the  circuit  subtends  an  angle  GO,  the  work  performed  is 


GO  being  the  solid  angle  subtended  by  that  face  of  the  cur- 
rent which  attracts  a  positive  pole. 

This  expression  therefore  represents  the  relative  energy 
of  the  current  and  the  pole.  If  the  latter  were  a  unit  pole, 
the  work  would  be  —kica. 

141.  Magnetic  Potential  Due  to  a  Circuit.  Unit  of 
Current  Ampere's  Hypothesis  on  the  Nature  of  Mag- 
netism. —  It  will  be  observed  that  the  expression  —  ktGo  an- 
swers to  the  definition  of  potential  in  terms  of  work  (§  12). 


MAGNETIC  PHENOMENA    DUE    TO    CURRENTS.      1  93 

We  are  moreover  justified  to  define  the  magnetic  forces 
due  to  the  current  by  -a  potential,  since  the  work  accom- 
plished in  the  field  of  the  current  is  a  function  of  the  co- 
ordinates of  the  circuit  traversed  by  the  current,  and  of  the 
point  where  the  magnetic  pole;t  is  supposed  to  be  placed. 
The  expression  —kioD  can  therefore  be  called  the  magnetic 
potential  due  to  the  current  at  the  given  point  where  the 
unit  pole  is  placed  : 

U=  -kioo.  (i) 

On  comparing  the  potential  —kioo  due  to  the  current 
with  the  potential  —F.oo  due  to  a  magnetic  shell  (§  43),  we 
recognize  the  identity  of  the  two  expressions.  A  current 
gives  the  same  potential,  and  consequently  produces  the 
same  magnetic  forces,  as  a  shell  of  the  same  outline,  whose 
strength  Fs  would  be  equal  to  ki.  To  determine  the  direc- 
tion of  the  forces  and  the  sign  of  the  potential,  it  is  neces- 
sary to  distinguish  the  two  faces  of  the  circuit.  According 
to  Ampere's  rule  (§  135),  the  face  of  the  circuit  which  pro- 
duces the  same  action  as  the  negative  side  of  a  shell  (that 
is,  attracts  a  north  pole)  is  that  around  which  the  current 
seems  to  go  in  the  direction  of  the  hands  of  a  watch  :  this  is 
the  negative  or  5  face  of  the  current,  the  other  is  the  posi- 
tive or  N  face. 

The  numerical  coefficient  k  of  equation  (i)  depends  on  the 
unit  chosen  to  measure  the  current.  It  could  be  put  equal 
to  unity,  and  the  unit  current  defined  as  that  current  which 
produces  unit  magnetic  potential  at  a  point  where  the  cir- 
cuit subtends  unit  solid  angle.  The  unit  thus  chosen  has 
the  same  dimensions  as  the  unit  of  strength  of  a  shell 


This  is  the  unit  which  we  shall  adopt  in  future  to  express  the 


194  ELECTROMAGNETISM. 

current ;  we  will  later  on  come  across  more  tangible  defini- 
tions of  it. 

Ampere,  who  discovered  the  identity  of  effect  of  a  shell 
and  a  current,  interpreted  this  identity  in  the  following 
manner  :  Experiment  shows  that  a  small  closed  current  acts 
like  a  small  magnet  normal  to  the  plane  of  the  current  on 
condition  that  the  moment  of  the  magnet  be  equal  to  the 
current  multiplied  by  the  surface  of  the  circuit.  Suppose 
any  finite  circuit  divided  by  lines  drawn  across  it  interiorly 
in  the  form  of  a  network  with  an  infinite  number  of  meshes, 
and  suppose  that  the  edges  of  these  meshes  are  traversed 
by  equal  currents  in  the  same  direction.  The  result  will  be 
currents  in  the  interior  lines  of  the  network  which  annul 
each  other  in  pairs ;  the  outer  contour  of  the  circuit  will  be 
the  only  seat  of  a  current.  Replacing  each  mesh  by  an 
equivalent  elementary  magnet,  the  combination  of  these  ele- 
ments forms  a  magnetic  shell  whose  effect  is  identical  with 
that  of  the  current  having  the  same  contour. 

From  the  preceding  comparison  Ampere  deduced  an  hy- 
pothesis which  refers  magnetic  phenomena  to  electric 
phenomena.  It  need  only  be  admitted  that  each  atom  of  a 
magnet  is  the  seat  of  a  circular  current ;  the  orientation  of 
these  currents  will  produce  effects  identical  with  those  of 
magnets.  We  must  then  adopt  as  a  postulate  that  such  an 
elementary  current  can  exist  without  expenditure  of  work, 
that  is,  that  electric  resistance  is  only  manifested  in  travers- 
ing interatomic  spaces.* 

Fleming  and  Dewar  have  recently  advanced  an  experi- 
mental proof  in  favor  ofth  is  latter  hypothesis.  They  showed 
that  if  the  variations  of  the  resistance  of  pure  metals  are 
represented  graphically  in  terms  of  their  temperatures, 
curves  are  obtained  which  appear  to  converge  toward  a 
point  of  no  resistance,  or  absolute  zero. 

*  See  Ampere,  Alemoires  public's  pur  la  Societd  de  Physique. 


MAGNETIC  PHENOMENA   DUE    TO   CURRENTS.      1  95 

There  is,  however,  a  distinction  to  be  established  between 
the  potential  due  to  a  current  and  that  given  by  a  shell 
Suppose  a  positive  magnetic  mass  equal  to  unity  be  placed 
against  the  positive  face  of  a  fetiell.  It  will  be  repelled,  will 
follow  a  curved  trajectory,  called  line  of  force,  and  will  bring 
up  against  the  negative  face,  where  it  will  remain  in  stable 
equilibrium.  The  work  accomplished  during  this  revolution 
is  471  Fs  (§  44). 

In  the  case  of  a  current,  the  N  pole  will  also  follow  a  line 
of  force,  but  as  this  latter  is  a  continuous  curve,  the  pole 
will  continue  to  move  in  this  orbit  as  long  as  the  current 
lasts.  Each  revolution  will  increase  the  work  done  by  the 
circuit  by  4?n,  and  consequently,  by  definition,  the  po- 
tential will  be  expressed  by 

U  =  —  t(a> 

n  denoting  the  number  of  revolutions  described  by  the  unit 
pole.  If  the  work  has  been  performed  by  the  magnetic 
force  due  to  the  current,  we  must  take  the  sign  -f,  for  the 
potential  will  then  have  decreased  ;  in  the  contrary  case,  that 
is  when  the  pole  has  been  forced  to  move  backwards,  the  sign 
—  must  be  chosen. 

It  follows  from  the  preceding  that  the  magnetic  potential 
due  to  a  current  contains  one  constant  more  than  the  po- 
tential of  a  shell.  Nevertheless,  as  regards  the  determination 
of  the  forces,  this  constant  is  eliminated,  since  the  intensity 
of  the  current's  magnetic  field  is,  in  a  direction  /, 


We  can  therefore  say  that  as  regards  exterior  magnetic 
actions  a  current  i  is  comparable  to  a  shell  F8  having  the 
same  contour. 


1 96  ELECT  ROM  A  ONE  TISM. 

The  total  magnetic  flux  produced  by  the  current  is  the 
sum  of  the  terms  of  the  same  sign,  such  as  JCd^,  which  can 
be  formed  in  an  equipotential  surface. 

The  electromagnetic  action  of  a  current  can  be  summed 
up  by  saying  that  the  current  magnetizes  the  medium  which 
surrounds  it,  and  develops  a  flux  of  magnetic  force  propor- 
tional to  the  intensity  of  the  electric  flux  and  the  permeabil- 
ity of  the  medium.  A  current  surrounded  with  iron  will  pro- 
duce a  very  much  greater  flux  than  if  it  were  surrounded  by 
air  or  some  feebly  magnetic  substance. 

142.  Energy  of  a  Current  in  a  Magnetic  Field.  Max- 
well's Rule. — Following  up  the  comparison  between  shells 
and  electric  circuits,  and  extending  the  expression  — imco 
which  has  been  found  for  the  relative  energy  of  a  current 
and  a  pole,  it  is  easy  to  see  that  the  relative  energy  of  a  cur« 
rent  and  a  field  is 

W=  -i$, 

0  denoting  the  flux  of  force  across  the  negative  face  of  the 
circuit. 

If  the  current  is  displaced  in  the  field,  the  work  accom- 
plished is  measured  by  the  variation  of  potential  energy. 
When  this  latter  becomes  a  minimum  the  circuit  reaches  a 
position  of  stable  equilibrium,  which  corresponds  to  a  max- 
imum flux  of  force  penetrating  across  the  negative  face  of 
the  current.  Hence  Maxwell's  rule: 

A  current  free  to  move  in  a  magnetic  field  tends  to  place  it- 
self so  as  to  receive  the  greatest  possible  flux  of  force  across  its 
negative  face. 

Thus,  a  circular  current  movable  about  one  of  its  diame- 
ters which  is  perpendicular  to  the  direction  of  the  earth's 
field  turns  so  as  to  point  its  positive  face  towards  the  north  : 
the  lines  of.  force  then  penetrate  perpendicularly  by  its  neg- 


MAGNETIC  PHENOMENA    DUE    TO    CURRENTS. 

ative  face.  If,  at  starting,  the  flux  entered  by  the  positive 
face  of  the  circuit,  this  movement  would  at  first  have  the 
result  of  reducing  the  number  of  lines  traversing  it ;  then 
towards  the  end  of  the  movement  the  flux  would  enter  by 
the  negative  face. 

143.  Relative  Energy  of  Two  Currents. — To  complete 
the  identity  of  currents  and  shells,  the  relative  energy  of  two 
circuits  traversed  by  currents  t,  i'  must  be  expressed  by 

W=-ii'Lm,        §46. 

The  factor  Lm  has  the  dimensions  of  a  length  and  is  called 
the  coefficient  of  mutual  induction  of  the  two  circuits.  By 
definition  (§  46),  Lmi  is  the  flux  sent  by  the  current  i  across 
i',  and  Lmir  the  flux  sent  by  i'  across  i. 

This  last  deduction  from  the  properties  of  shells  was  not 
a  priori  evident,  for  it  does  not  necessarily  follow,  from  the 
fact  that  two  currents  act  upon  a  pole,  that  they  act  upon 
each  other;  e.g.,  two  pieces  of  soft  iron  act  on  a  magnet, 
but  taken  separately  they  have  no  influence  upon  each  other. 
It  was  Ampere  that  discovered  the  existence  of  the  forces, 
called  electrodynamic,  exercised  between  currents. 

144.  Intrinsic  Energy  of  a  Current. — A  circuit  carrying 
a  current  is  traversed  by  the  lines  of  force  which  it  gener- 
ates, and  which  form   closed  curves  around  the  conductor. 
The  figure  assumed  by  these  lines  in  a  plane  can  be  shown 
by  the  use  of  iron-filings  strewn  on  a  sheet  of  paper  through 
which  the  current  passes.     This  figure  is  analogous  to  that 
fora  lamellar  magnet  with  the  same  contour,  and  magnetized 
on  its  opposite  faces. 

Suppose  the  circuit  be  placed  in  a  medium  of  constant 
permeability,  for  example  air,  and  denote  by  Lt  the  flux  of 


1  98  ELECT  ROM  A  GNE  TISM. 

force  passing  in  the  circuit  when   the  current   is  equal  to 
unity.     For  a  current  c  the  flux  will  be 

Lsi  =  0. 

Now  a  current  traversed  by  a  flux  possesses  a  reserve  of 
potential  energy,  the  variation  of  which  measures  the  work 
performed.  In  the  case  we  are  considering,  the  flux  is  de- 
pendent on  the  current  ;  in  order  to  find  the  expression  for 
the  energy  it  is  necessary  to  use  the  method  of  reasoning 
employed  in  regard  to  the  phenomena  of  electrification  or 
magnetization  (§  25). 

When  the  current  varies  by  dz,  the  flux  varies  by  d#,  and 
the  potential  energy  by 


This  energy  is  essentially  positive,  for  the  setting  up  of 
the  current  demands  an  expenditure  of  energy. 

If,  then,  the  current  passes  from  o  to  c,  the  energy,  which 
at  first  is  zero,  becomes 


or 


This  expression  represents  the  intrinsic  energy  of  the  cur- 
rent. 

The  coefficient  Ls,  whosed  imensions  like  those  of  the  co- 
efficient of  mutual  induction  reduce  to  a  length,  is  called  the 
self  -inductance  )  or  coefficient  of  self-induction,  of  the  circuit. 

145.  Faraday's  Rule.—  Before  passing  to  the  applications 
of  these  various  formulae,  let  us  try  to  find  another  expres- 


MAGNETIC  PHENOMENA    DUE    TO    CURRENTS. 

sion  besides  Maxwell's  (§  140)  for  the  work  done  in  displace- 
ments of  a  circuit  in  a  field.  Maxwell  considers  the  circuit 
as  a  whole,  and  includes  in  one  simple  formula  the  work 
done  in  a  deformation  or  a  displacement  of  the  conductors. 

It  is  often  useful  to  analyze  Separately  the  action  of  the 
various  parts  of  a  circuit,  and  to  determine  the  part  which 
belongs  to  each  one  of  them  in  the  work  accomplisned. 

For  this  purpose  let  us  consider  again  the  expression  for 
the  work  of  an  element  of  current  ds  (§  139),  which  is  dis- 

rH, 

placed  by  ds'  in  a  field  of  intensity  3C  =  -^-,  due  to  a  pole 
m, 

dW  =  iWds  sin  (/, ds)ds'  cos  (d/, ds'). 
Now  the  product 

OCdj  sin  (/,  ds)dsf  cos  (df,ds')> 

which  represents  the  product  of  the  field-intensity  by  the 
projection  of  the  area  described  by  the  given  current-ele- 
ment upon  a  plane  normal  to  the  direction  of  the  field,  is 
simply  the  flux  of  force  swept  over  by  the  conductor,  for  JC 
is  the  flux  per  unit  equipotential  surface. 

It  is  easy  to  extend  this  to  a  conductor  of  finite  length 
and  to  obtain  the  following  rule,  first  pointed  out  by  Fara- 
day : 

The  ^vork  accomplished  by  a  conductor  which  is  displaced  in 
a  field  is  equal  to  the  product  of  the  current  by  the  flux  of 
force  (or  number  of  lines  of  force]  cut  by  the  conductor. 

It  should  be  further  remembered  that  the  current  tends  to 
move  towards  the  left-hand  of  Ampere's  manikin  when  he 
faces  in  the  direction  of  the  field.  If  the  conductor  is 
moved  in  such  a  way  as  to  cut  no  lines  of  force,  the  work 
accomplished  is  zero.  This  is  the  case  when  the  conductor 
is  displaced  parallel  to  the  direction  of  the  field. 


200  ELECTROMAGNETISM. 


APPLICATIONS    RELATING    TO    THE    MAGNETIC    POTENTIAL 
OF  THE   CURRENT. 

146.  Case  of  an  Indefinite  Rectilinear  Current— Let  us 
see  whether  the  application  of  the  idea  of  potential  leads  to 
the  expression  for  the  electromagnetic  force  found  by  Biot 
and  Savart  (§  136),  in  the  case  of  an  indefinite  rectilinear 
current  acting  on  a  neighboring  pole. 


FIG.  67. 

Such  a  current  projected  on  O  can  be  considered  as  the 
limit  of  a  plane  circuit  projected  along  OO't  and  extended  in- 
definitely towards  the  right.  The  conductors  which  com- 
plete the  circuit  being  infinitely  removed  from  the  pole, 
supposed  to  be  at  P,  have  no  action  upon  it. 

Let  us  now  interpret  the  expression  for  the  potential 

U  —  —  i(a> 

The  solid  angle  GO,  subtended  by  the  circuit  at  the  point 
P,  is  obtained  by  cutting  the  sphere  of  unit  radius,  drawn 
about  P,  by  a  diametral  plane  OP,  which  includes  all  the 
right  lines  joining  the  point/*  to  the  conductor  projected 
on  O,  and  by  a  second  plane  PP'9  which  likewise  includes 


MAGNETIC  POTENTIAL   OF   THE    CURRENT.        2O1 

all  the  lines  drawn  from  P  'to  the  infinitely  distant  limit  of 
the  given  imaginary  circuit.     The  portion  of  the  spherical 
surface  thus  cut  off  is  a  segment  which  is  measured  by  twice 
the  dihedral  angle  a  between,  the  planes  OP,  PP. 
We  have  therefore  ^ 

U=  —i(±  2a 

the  sign  of  2a  being  positive  or  negative  according  as  the 
current  flows  upwards  or  downwards. 

The  potential  has  consequently  a  constant  value  in  the 
plane  OP,  which  is  equipotential. 

The  intensity  of  the  magnetic  field  produced  by  the  cur- 
rent is 


- 

ds 

In  any  point  of  the  plane  OP  the  forces  of  the  field  are 
directed  perpendicularly  to  this  plane  ;  let  us  now  try  to  find 
the  value  of  the  intensity  in  this  direction. 

Let 


we  have 

ds  —  ld(n  —  a)  =  —  /da, 
whence 


This  expression  conforms  with  Biot  and  Savart's  law.  If 
the  current  O  flows  upwards,  P  tends  to  approach  the  shell 
and  the  sign  of  the  force  is  negative  ;  the  sign  is  positive  in 
the  contrary  case. 

The  dotted  circumferences  around  the  point  O  show  lines 
of  force  which  correspond  to  field-intensities  decreasing  in 
geometrical  progression.  The  planes  passing  through  the 


202 


ELECT  ROM  A  GNE  TISM. 


axis  of  the  conductor  are  normal  to  the  lines  of  force,  and 
consequently  equipotential. 

147.  Case  of  a  Circular  Current.  Tangent-galvanom- 
eter.— A  circular  current  of  radius  R  subtends,  at  a  point 
P  on  its  axis  OP,  a  solid  angle  GO  measured  by  the  spherical 
segment 

+  27i(\  —  cos  a). 

If,  for  an  observer  situated  at  P,  the  movement  of  the  cur- 
rent is  in  the  same  direction  as  the  hands  of  a  watch,  the 
potential  at  the  point  P  is 

U  =  —  i(a>  ±  47rn)  =  —  2ni(i  —  cos  a  ±  2n). 
ffk 


FIG.  68. 


The  intensity  of  the  field  due  to  the  current  must  be  di- 
rected along  the  axis  OP  by  reason  of  symmetry;  it  is 
therefore  expressed  by 


dr 


(r*  +  *•)! 

That  is,  on  the  above  hypothesis,  P  is  attracted  towards  O. 
If  the  point  />were  at  the  centre  of  the  circle,  the  intensity 
would  become 

Ic 


I  being  the  length  of  the  current. 


MAGNETIC  POTENl^IAL   OF   THE    CURRENT.         203 

If  there  were  n  circular  currents,  so  near  together  that 
their  mutual  distances  were  negligible  compared  to  the  ra- 
dius R,  the  intensity  at  the  centre  would  be 


Fig.  69  shows  the  distribution  of  equipotential  lines  and 
lines  of  force  (marked  by  arrows)  in  a  field  due  to  a  circular 


FIG.  69. 


FIG.  70. 


current.  The  intensity  varies  in  inverse  ratio  to  the  dis- 
tance between  the  equipotential  lines,  and  directly  as  the 
density  of  the  lines  of  force. 

Suppose  a    magnet-needle   of  very  small  dimensions  be 


204  ELECTROMAGNETISM. 

hung  by  a  silk  fibre  of  negligible  torsion  in  the  centre  of  a 
vertical  circular  frame,  round  which  are  wound  n  spiral  coils 
of  wire,  very  close  together.  Suppose,  moreover,  that  the 
frame  be  oriented  in  the  plane  of  the  magnetic  meridian. 

When  a  current  i  is  sent  through  the  spirals,  the  needle  is 
acted  on  by  the  electromagnetic  force,  on  the  one  hand, 
which  tends  to  put  it  crossways  to  the  current;  on  the  other 
hand,  by  the  terrestrial  magnetism,  the  action  of  which  op- 
poses this  movement. 

Under  the  influence  of  these  contrary  actions,  the  needle 
takes  up  a  position  of  equilibrium  corresponding  to  an  angle 
a  with  the  meridian. 

Denoting  by  2fft  the  magnetic  moment  of  the  needle,  by 
5C,  the  horizontal  component  of  the  earth's  field,  the  couple 
due  to  the  earth  is 

3710C  sin  a     (§  37). 
The  couple  due  to  the  current  is 

SfliaC"  cos  a  =  371  —£-  cos  a. 
JK. 

As  these  two  couples  are  in  equilibrium, 
2nni 


whence 


cos  a  —  X  sin  a, 
K 


RW, 

= tan  a. 

2nn 


Knowing  3C,  R,  and  n,  and  measuring  a  by  one  of  the 
methods  shown  in  §  50,  we  can  deduce  from  the  preceding 
-expression  the  strength  of  the  current  around  the  frame. 
This  apparatus  is  called  the  tangent-galvanometer.  The 
spirals  of  wire  are  called  the  galvanometer-coil  or  multi- 


MAGNETIC  POTENTIAL    OF   THE   CURRENT.         2O$ 
r> 

plier;  the  quantity is  the  reduction  factor  of  the  galvano- 
meter. 

148.  Thomson  Galvanometers.— The  application  of  the 
above  simple  formula  "necessitates  such  a  displacement  be- 
tween the  coil  and  the  poles  of  the  needle  that  the  tangent- 
galvanometer  is  by  no  means  sensitive. 

In  order  to  measure  weak  currents,  the  multiplier  must 
be  wound  very  close  to  the  needle.  In  order  to  determine 
the  best  form  to  give  it  with  a  view  to  economize  the  wire 
and  diminish  the  resistance,  let  us  take  up  again  the  expres- 
sion for  the  action  of  a  circular  current  on  unit  pole  situated 
in  0,  Fig.  71. 


OC  —  2ni 


(r>  + 


If  we  put  3C  =  const.,  and  consider  r  and  R  as  variable, 
we  get  the  equation  of  a  curve  having  two  symmetrical  parts 
with  regard  to  o,  Fig.  71. 


FIG.  71. 

The  area  limited  by  this  curve  is  the  meridian  section  of 
a  volume  of  revolution  about  which  the  wire  can  be  wound. 
All  the  spirals  composing  such  a  volume  will  have  an  action 
on  the  unit  pole  at  least  equal  to  JC.  All  the  spirals  exterior 


2O6 


ELECTROMA  GNE  TISM. 


to  this  volume  will  have  a  less  action.  The  cross-hatched 
section  (Fig.  59)  which  allows  for  a  cylindrical  cavity  in 
which  to  place  the  needle  represents,  therefore,  the  rational 
form  for  the  coil  of  a  very  sensitive  galvanometer.  Lord 
Kelvin  has  approached  as  nearly  as  possible  to  this  form  in 
the  construction  of  his  galvanometers. 


In  order  to  increase  the  sensitiveness,  it  is  possible,  in  addi- 
tion, to  diminish  the  action  of  the  earth  on  the  needle.  Two 
means  are  employed  for  this  purpose.  The  rod  which  sup- 
ports the  needle  and  passes  through  the  coil  carries  a  second 
needle,  »Y,  oriented  in  the  opposite  direction  to  the  first 
(Fig.  72).  By  choosing  two  needles  of  slightly  different 
magnetic  moments,  the  directive  couple  due  to  the  earth 
can  be  diminished  as  much  as  desired.  Such  a  system  of 
of  needles  is  called  astatic.  The  earth's  action  may  likewise 
be  modified  by  a  compensating  magnet  NS,  placed  above 
the  needle.  By  changing  the  position  and  orientation  of 
this  magnet  it  is  possible  to  diminish  or  increase  its  compen- 
sating, or  directive,  effect  at  will.  These  various  methods 
are  often  employed  together  in  Lord  Kelvin's  galvanometers. 

In  consequence  of  the  complex  form  of  the  coil  and  the 
nearness  of  the  current  to  the  poles  of  the  magnet,  the  cur- 


MAGNETIC  POTENTIAL    OF   THE   CURRENT.         2O/ 

rent  c  is  not  related  to  the  deviation  a  by  a  simple  formula, 
as  in  the  case  of  the  tangent-galvanometer. 

If  i  is  developed  as  a  function  of  a,  according  to  Mac- 
laurin's  series,  it  will  take  the  fprm 


The  function  must  become  zero  when  a  is  zero,  whence 
it  follows  that 

S(o)  =  o. 

Moreover,  if  the  deviation  is  very  small,  we  can  neglect 
the  third  and  subsequent  terms,  and  take  the  current  as  pro- 
portional to  the  deviation 

i  =  ka. 

The  coefficient  k,  which  is  the  reduction  factor,  is  deter- 
mined by  sending  a  current  of  known  strength  through  the 
coils. 

The  last  formula  is  admissible  for  deviations  of  less  than 
3°  ;  the  readings  are  made,  necessarily,  by  the  method  of 
reflection  (§  50). 

149.  Shunt—  When  the  deviation  exceeds  this  limit,  it 
is  reduced  by  an  artificial  resistance  called  a  shunt  placed 
around  the  galvanometer. 

The  partial  current  through  galvanometer  is  then,  de- 
noting by  g  the  resistance  of  the  apparatus,  and  by  s  that 
of  the  shunt  (§  121), 


whence 


=  g—-=  mg. 


208  ELECTROMA  GNE  TISM. 

The  factor  m  is  called  the  multiplying  power  of  the  shunt. 
It  is  the  factor  by  which  the  value  of  the  current  as  found 
by  the  galvanometer  must  be  multiplied,  in  order  to  obtain 
the  total  current. 

150.  Measurement  of  an  Instantaneous  Discharge.  — 

Suppose  a  quantity  of  electricity  q  traverses  the  coil  of  a 
tangent-galvanometer  with  such  rapidity  that  the  needle  is 
not  displaced  by  an  appreciable  quantity  during  the  dis- 
charge. Suppose,  too,  that  the  movement  of  the  needle  is 
not  damped,  so  that  a  double  oscillation  is  obtained  having 
a  duration 

"'"  (§37)    •    •    -d) 

Let  us  express  the  fact  that  the  kinetic  energy  of  the 
needle  is  equal  to  the  terrestrial  couple,  and  that  its  momen- 
tum represents  the  impulse  that  has  been  given  to  it. 

In  expressing  these  facts  it  must  be  remembered  that 
equations  relative  to  movements  of  translation  are  appli- 
cable to  movements  of  rotation,  if  the  masses  be  replaced 
by  the  moments  of  inertia,  the  forces  by  the  couples,  and  the 
linear  by  the  angular  velocities. 

Let  GO  be  the  initial  angular  velocity,  and  a  the  maxi- 
mum deviation  of  the  needle.  The  terrestrial  couple  is,  for 
an  angle  «, 


sn  a. 
The  equation  of  the  kinetic  energies  gives 


—  =  J  'sfiwe  sin  ad<x  =  9fTWe(i  —  cos  a 


=  23710C  sin      i.  .      . (2) 


MAGNETIC  POTENTIAL    OF   THE    CURRENT.         2OQ 

Let  i  be  the   current  of  discharge ;    its   action   on    the 
needle  produces  a  couple 


The  impulse  communicated  to  the  needle  is 

/  =  (2mr>)oj,  ....     (3) 


T  representing  the  duration  of  the  discharge. 
Now 


fi 
t/ 


idt 


is  the  quantity  of  electricity  q  which  was  discharged. 

Eliminating  ^mr*  and  09  between  equations  (i),  (2),  and 

(3),  we  get 


T  LW    .     a, 

q  =  --  -  sin  —  -• 
n  2nn          2 


In  the  case  of  a  very  small  deviation  we  get  simply 


T  , 

q  —  --      —  =  Aa 
7t  2nn  2 


The  quantity  of  electricity  is  then  proportional  to  the  arc 
of  the  needle's  swing.  Under  the  "condition  that  the  devia- 
tions are  slight,  this  formula  may  be  extended  to  any  form  of 
mirror-galvanometer  whatever. 

In  order  to  satisfy  the  condition  at  the  beginning  of  this 
paragraph,  galvanometers  are  chosen  for  this  purpose  with 
a  heavy  needle  having  considerable  moment  of  inertia. 
These  needles  swing  slowly,  and  enable  us  to  read  the  limit 
of  the  swing  exactly. 


2  1  0  ELECTROMA  GNE  7  'ISM. 

151.  Solenoid.  Cylindrical  Bobbin.—  Under  the  name 
of  solenoid  Ampere  defined  a  series  of  equal  circular  cur- 
rents, placed  very  close  together  and  normal  to  a  rectilinear 
or  curvilinear  axis  passing  through  the  centres  of  gravity  of 
the  surfaces  which  have  these  currents  as  their  edges. 

Denote  by  s  the  surface  of  the  circuits,  by  e  their  distance 
apart,  and  by  i  the  current.  Each  of  them  can  be  replaced 
by  a  shell  of  the  same  contour,  and  having  a  strength  i. 
We  may  choose  the  thickness  of  the  shells  arbitrarily:  we 
will  take  it  equal  to  e.  Denoting  by  a  the  magnetic  density 
of  the  faces  of  the  shells,  and  observing  that 

ea  =  i, 
we  have 


nl  representing  the  number  of  currents  per  unit  length. 

The  adjoining  faces  of  neighboring  shells  counterbalance 
each  other,  and  these  remain,  at  the  extremities  of  the 
series,  poles  whose  mass  is 


m  = 


A  bobbin  formed  on  a  layer  of  insulated  wire  wrapped 
round  a  cylindrical  core  may  be  considered  as  a  solenoid 
when  the  wire  is  traversed  by  a  current.  As  a  result,  how- 
ever, of  the  obliquity  of  the  spirals  of  the  bobbin  there  is 
an  exterior  action,  which  may  be  obviated  by  bringing  back 
the  ends  of  the  wire  along  the  axis  of  the  bobbin  (Fig.  73). 

If  there  are  an  even  number  of  layers  of  wire  on  the  core 
with  their  spirals  inclined  in  opposite  directions,  the  effects 
due  to  their  obliquity  are  rendered  null. 

Supposing  the  thickness   of  the  layers   to  be   negligible 


MAGNETIC  POTENTIAL    Of    THE   CURRENT.         211 

compared  to  their  diameter,  the  resultant  poles  of  the  bob- 
bin are  given  by 

m  = 


nl  being  the  number  of  turns  p§r  unit  length,  and  s  their 
mean  surface.     The  magnetic  moment  of  the  solenoid  is 

ml  =  ins, 
n  representing  the  total  number  of  turns  of  wire 


Such  a  solenoid  has  all  the  magnetic  properties  of  a  uni- 
form cylindrical  magnet  (§  42). 

The  earth's  action  on  the  solenoid  is  shown  by  hanging  it 
from  the  ends  of  the  wire  (plunged  into  cups  of  mercury), 
which  serve  as  pivots,  and  at  the  same  time  give  access  to 
the  current  from  a  battery,  Fig.  73.  By  this  means  it  is 
shown  that  the  positive  face  of  the  solenoid  turns  towards 
the  north. 

To  determine  the  intensity  of  the  field  due  to  the  sole- 
noid, it  need  only  be  remembered  that  it  can  be  replaced 
by  a  uniform  cylindrical  magnet. 

Its  action  on  unit  pole  situated  on  its  axis,  at  some  ex- 
ternal point,  is 

5C  =  o-(2oo  —  a/),         (§  31) 


212  ELECTROMAGNETISM. 

GO  and  GO'  being  the  solid  angles  subtended  by  the  bases  of 
the  cylinder  at  the  given  point. 

If  the  unit  pole  is  situated  in  the  plane  of  one  of  the 
bases,  the  above  equation  becomes 

OC  =  a(27t  —  GO'). 

Finally,  in  order  to  obtain  the  expression  for  the  field  at 
a  point  inside  the  solenoid,  suppose  the  latter  to  be  cut  into 
two  portions  by  a  plane  through  the  given  point.  The  total 
action  is  the  sum  of  the  actions  of  the  two  portions.  Now 
the  effect  of  the  first  is 

<j(27T  —  6?), 

that  of  the  second 

<r(27r  =  GO'). 

As  these  actions  are  added  the  resultant  action  is 
OC  —  a(2n  —  GO)  +  a(2n  —  GO'). 

In  the  particular  case  where  the  cylinder  extends  a  con- 
siderable distance  on  each  side  from  the  point  the  angles 
GO  and  GO'  become  negligible  compared  to  2n,  and  the  action 
is  expressed  by 

3C  =    7TG-  = 


This  expression  represents  the  flux  per  unit  of  section 
taken  normally.     The  total  flux  is 


This  flux  remains  constant  in  the  cylinder  up  to  a  certain 
distance  from  the  ends.  When  near  the  ends  the  intensity 
of  the  field  decreases,  since  one  of  the  solid  angles  GO  or  GO' 
assumes  an  increasing  value.  The  internal  flux  conse- 


MAGNETIC  POTENTIAL    OF   THE   CURRENT.         21$ 

quently  decreases,  and,  as  in  the  case  of  a  magnet,  lines  of 
force  make  their  exit  through  the  sides  of  the  cylinder. 
The  total  flux  A^nn^is  is  consequently  divided  into  two  parts, 
one  emerging  from  the  cylinder  by  the  positive  face,  the 
other  by  the  sides.  The§e  two  s£ts  of  lines  of  force  spread 
out  through  the  surrounding  space,  and  re-enter  in  the  same 
way  by  the  negative  face  of  the  cylinder  and  the  adjoining 
lateral  walls. 

We  conclude  from  the  above,  that  inside  a  cylindrical 
bobbin  of  great  length  a  uniform  magnetic  field  is  produced, 
directed  parallel  to  the  axis  of  the  cylinder  from  the  5  to 
the  A^ace  ;  the  intensity  of  the  field  is  measured  by  47r,  mul- 
tiplied by  the  product  of  the  current  into  the  number  of 
spirals  turns  in  unit  length. 

Such  a  bobbin  consequently  furnishes  a  practical  means 
of  obtaining  a  uniform  field  whose  intensity  is  only  limited 
by  the  heating  of  the  wire  by  the  current. 

The  identity  of  the  external  effects  of  solenoids  and  mag- 
nets has  led  to  the  conclusion  that  there  is  an  analogy  in 
their  internal  effects,  which  latter  cannot  be  directly  deter- 
mined in  the  case  of  magnets.  It  is  known,  indeed,  that  if 
a  cavity  be  made  in  a  magnet,  its  walls  form  poles  whose 
effect  is  added  to  that  of  the  end-poles.  Care  must  be 
taken,  however,  not  to  confuse  the  internal  action  of  a  sole- 
noid with  that  of  a  tubular  magnet,  for  in  such  a  magnet  the 
lines  of  force  have  the  same  direction  inside  and  outside, 
their  return  taking  place  in  the  thickness  of  the  tube. 

To  sum  up,  it  is  admitted  that  magnets,  like  solenoids, 
gives  a  constant  and  continuous  total  flux,  which  makes  its 
exit  by  the  N  end,  and  returns  to  the  point  of  departure 
by  entering  through  the  5  end.  We  shall  see  that  this  con- 
sideration has  led  to  a  comparison  between  magnetic  and 
electric  fluxes,  and  to  treating  the  former  by  relations 
analogous  to  the  latter. 


2 1 4  ELECTR  OMA  GNE  TISM. 

152.  Electrodynamometer. — The  magnetic  properties  of 
solenoids  have  been  carefully  verified  by  Weber,  who  by 
means  of  them  has  repeated  Gauss'  experiments  on  mag- 
netism (§  48).  To  give  the  desired  mobility  to  one  of  the 
solenoids,  it  is  hung  by  two  slender  wires,  which  at  the  same 
time  serve  for  the  entrance  and  exit  of  the  current. 

The  solenoid  may  be  supported  by  two  wires  placed  side 
by  side:  in  this  case  the  torsion-couple  which  balances  the 
mutual  action  of  the  two  solenoids  is  proportional  to  the 
sine  of  the  angle  of  torsion.  If  the  suspension  wires  are 
one  above  and  one  beneath  the  solenoid  in  the  same  right 
line,  the  moment  of  torsion  is  simply  proportional  to  the 
angle  of  torsion,  and  one  of  the  wires  supports  the  whole 
weight  of  the  movable  solenoid. 

The  mutual  action  of  the  two  solenoids  is  proportional  to 
their  magnetic  moments,  and,  consequently,  to  the  product 
of  the  currents  which  traverse  them.  If  the  same  current 
passes  through  both,  the  couple  is  proportional  to  its 
square. 

Weber  has  utilized  these  properties  in  the  construction 
of  the  electrodynamometer.  In  this  apparatus,  which  serves 
to  measure  the  current-strength,  the  movable  solenoid  is 
hung  at  the  centre  of  the  fixed  one  and  at  right  angles  with 
it. 

The  current  to  be  measured  passes  successively  through 
both  solenoids,  whose  axes  are  then  urged  to  assume  a 
parallel  position.  The  turning  couple,  balanced  by  the  tor- 
sion of  the  suspension  wires,  is  proportional  to  the  square 
of  the  current.  This  property  allows  the  apparatus  to  be 
applied  to  the  measurement  of  currents  whose  flux  varies 
periodically  in  direction,  since  the  mutual  action  of  the  two 
solenoids  preserves  the  same  sign,  whatever  be  the  sign  of 
the  current. 

The   action    of   terrestrial    magnetism    on    the    movable 


MAGNETIC  POTENTIAL    OF   THE    CURRENT. 


215 


solenoid  must  be  taken  into  account.  The  solenoid  can, 
however,  be  reduced  to  a  very  small  number  of  spirals,  so  as 
to  render  this  effect  of  but  little  account. 

A  more  exact  method  consists  In  first  adjusting  the  mov- 
able solenoid  so  that  its  axis  is^jn  the  magnetic  meridian, 
with  the  positive  end  turned  towards  the  north.  When  the 
solenoid  is  deflected  under  the  influence  of  a  constant  cur- 
rent, it  is  brought  back  to  its  initial  position  by  turning  the 
upper  part  of  the  suspension  wire.  The  angle  of  torsion 
enables  us  to  measure  the  deflecting  current,  the  terrestrial 
couple  being  zero. 

The  angle  of  torsion  /?  is  proportional  to  the  square  of 
the  current  through  the  two  solenoids,  or 

ft  =  ki\ 

153.  Case  of  a  Ring-shaped  Bobbin  or  Solenoid.— Sup- 
pose a  layer  of  wire  wound  so  as  to  form  a  ring  of  rectangu- 
lar section  (Fig.  74),  each  spiral  being  situated  in  a  meridian 
section.  The  magnetic  effect  of  such  a  bobbin,  traversed 


FIG.  74. 

by  a  current,  is  zero  in  all  external  points.  The  internal 
flux  of  force  created  by  the  system  is  composed  of  lines  of 
force  concentric  with  the  ring.  This  system  is  equivalent 


2 1 6  ELECT R  OMA  GNE  TISM. 

to  a  magnet  formed  of  magnetic  filaments  closed  on  them- 
selves. 

The  field  is  variable  in  a  meridian  section  of  the  ring.  It 
is  easy  to  see,  indeed,  that  the  number  of  turns  of  wire  per 
unit  length  is  smaller  on  the  outside  than  on  the  inside  of 
the  ring  ;  it  follows  that  the  field  within  the  ring  is  more 
intense  at  the  inner  periphery  than  at  the  outer. 

In  order  to  determine  the  intensity  of  the  interior  field, 
denote  by  n'  the  number  of  spirals  comprised  between  two 
meridian  planes  whose  angular  opening  equals  one  radian. 
At  a  distance  r  from  the  axis  the  space  between  two  adjoin- 
ing spirals  is  e  —  — .  The  intensity  of  magnetization  of  the 
equivalent  magnetic  solenoid  is 


corresponding  to  this  intensity,  per  unit  surface,  is  a  flux 
equal  to 

n'i 

47T3  =  4?T  -- 

Across  an  element  of  section  ds,  at  a  distance  r  from  the 
axis,  the  flux  is 

,.ds 
i  — 

The  total  flux  across  a  section  of  the  ring  whose  height  is 
a,  thickness  #,  and  interior  radius  R,  is 


/••;?  +  b  /»/?  +  * 

.   /  ds  ,.   I  adr 

/  —  •  =  4nnt  I  --  = 
J  R  r  J  R  r 


Rb 
-= 
K 


MAGNETIC  POTENTIAL    OF   THE    CURRENT.         2 1/ 

Calling  n  the  total  number  of  spirals  and  nt  the  number 
per  unit  length  along  the  inside  face  of  the  ring,  we  have 


2n       | 

whence 

R-\-b 


.    .  .„    . 

=  2ma  log,  — - —  =  47tn1zRa  \oge 


The  mean  intensity  of  the  field  inside  the  ring  is 


_ 

»t  —  ~~r  — 
ab 


It  is  easily  proven  that  this  expression  becomes  47tnli  for 
a  ring  of  very  great  interior  diameter. 

In  fact,   developing  the  logarithm  in  series,  we  get 


If  b  is  negligible  compared  to  R,  we  get  simply 


In  the  case  of  a  ring  of  circular  section,  where  the  radius 
of  the  circular  axis  of  the  ring  is  R,  and  that  of  a  meridian 
section  a,  we  get 


R-a 


2 1 8  ELECTROMA  GNE  TJSM. 

Hence  a  mean  internal  intensity  equal  to 


oe»  = 


Developing  the  expression  under  the  radical  sign  into  a 
ries  and  neglecting  the  ratio  —  in 
great  diameter,  we  would  have  simply 


series  and  neglecting  the  ratio  —  in  the  case  of  a  ring  of 


KLECTRO  MAGNETIC   ROTATIONS  AND  DISPLACEMENTS. 

154.  An  invariable  system  of  electric  currents  is  not  cap- 
able of  producing  the  rotation  of  a  magnet,  for  if  the  mag- 
net returns  to  its  initial  position  after  having  traversed  a 
current  c,  the  work  accomplished  by  its  poles  of  masses 
+  m  and  —  m  is 

-f-  ^itmi  —  4nmi  =  o.         (§141) 


If  the  magnet  does  not  traverse  the  current,  the  work  is 
equally  zero,  so  that  in  no  case  can  the  resistance  due  to 
friction  be  overcome. 

But  rotation  can  be  produced  by  various  artifices,  such 
as  rendering  only  one  part  of  the  electric  or  magnetic  sys- 
tem movable,  changing  the  direction  of  the  current,  etc. 

155.  Rotation  of  a  Current  by  a  Magnet.—  A  vertical 
magnet  serves  as  pivot  to  a  balanced  conductor  whose 
lower  end  plunges  into  a  circular  trough  of  mercury  which 
extends  round  the  middle  of  the  magnet.  A  voltaic  ele- 
ment is  placed  in  the  circuit  between  the  trough  and  the 
magnet  so  as  to  furnish  a  steady  current.  According  to 


ROTATION  AND   DISPLACEMENTS. 


2IQ 


Faraday's  rule  (§    145),   the   movable   conductor    tends   to 
move  so  as  to  cut  the  lines  of  force  due  to  the  magnet. 
In  the  course  of  one  revolution  the  conductor  cuts  all  the 


FIG.  75. 

lines  of  force  emanating  from   one  of  the  poles  m,  or 
lines,  according  to  Gauss'  theorem.    The  work  accomplished 
is  therefore  (§  145) 

w  =  i  X 


The  couple  acting  on  the  conductor  is 


w 

27t 


156.  This  experiment  can  be  repeated  by  the  aid  of  an 
electric  discharge  in  the  form  of  a  brush-discharge,  obtained 


FIG.  76. 
by  maintaining  two  conductors,  a,  b,  at  a  considerable  differ- 


22O 


ELECTRO  MA  GNE  TISM. 


ence  of  potential  inside  a  vessel  from  which  the  air  is  par- 
tially exhausted.  If  the  conductor  a  passes  round  the  mid- 
dle of  the  magnet  ns  the  luminous  band  is  seen  to  turn  about 
the  magnet  by  reason  of  the  electromagnetic  force  which 
impels*  it. 

A  liquid  traversed  by  a  current  and  situated  in  a  mag- 
netic field  will  likewise  assume  a  movement  normal  to  the 
lines  of  force  of  the  field. 

157.  Barlow's  Wheel. — This  apparatus  gives  another 
example  of  the  rotation  of  a  current  under  the  influence 
of  a  magnet.  It  consists  of  a  copper  wheel  whose  metallic 


FIG.  77- 

axis  conducts  the  current  to  the  wheel,  from  which  it  passes 
into  a  trough  of  mercury,  into  which  the  lower  edge  of  the 
wheel  dips. 

The  part  of  the  wheel  traversed  by  the  current  is  em- 
braced between  the  poles  of  a  horseshoe  magnet,  N.  The 
electromagnetic  force  tends  to  turn  the  wheel  in  the  direc- 
tion of  the  arrow.  By  Faraday's  rule  (§  145),  the  work  per 
revolution  is 

w  = 


3C  denoting  the  mean  intensity  of  the  field  cut  by  the  wheel 
s  the  surface  described  by  one  of  its  radii — that  is,  the  sur- 
face of  the  wheel  itself,  and  c  the  current. 


ROTATION  AND   DISPLACEMENTS.  221 

158.  Rotation  produced  by  Reversing  a  Current— The 
various  preceding  examples  suppose  no  variation  in  the  direc- 
tion of  the  current  ;  in  the  following  case  this  reversal  is  the 
cause  of  rotation.  Suppose^  a  bobbin  movable  round  an 
axis  perpendicular  to  its  own,  in  a  field  which,  for  simplic- 
ity, we  will  suppose  uniform. 


FIG.  78, 

The  ends  of  the  insulated  conductor  which  is  wrapped  on 
the  bobbin  dip  into  a  trough  divided  by  a  partition  par- 
allel to  the  field  into  two  compartments  filled  with  mercury. 
If  a  current  is  passed  through  the  mercury  into  the  bobbin, 
the  latter  tends  to  orient  itself  so  as  to  embrace  the  greatest 
possible  flux  by  its  negative  face  (§  142).  But  at  the  mo- 
ment when  this  position  is  reached,  the  moving  contacts 
cross  the  partition  in  consequence  of  the  acquired  velocity; 
the  current  is  then  reversed  in  the  bobbin,  the  electromag- 
netic couple  preserves  the  same  sign,  and  the  movement 
continues  as  long  as  the  current. 

Denoting  by  s  the  mean  surface  of  the  spirals,  by  n  their 
number,  and  by  3C  the  field-intensity,  the  work  accomplished 
in  one  revolution  is  (§  142) 


X 


222  ELECTROMAGNET1SM. 

159.  Mutual  Action  of  Currents.—  The  form  of  the  field 
produced  by  a  current  being  given,  we  need  only  apply 
Ampere's  rule  (§135)  to  predict  the  reaction  of  two  cur- 
rents. It  is  readily  seen  that  two  parallel  currents  attract 
each  other  when  they  have  the  same  direction,  and  repel 
each  other  if  they  have  opposite  directions. 

In  consequence  of  this  the  spirals  of  a  solenoid  tend  to 
approach  each  other  when  they  are  the  seat  of  an  electric 
flux. 

Two  angular  currents  attract  or  repel  each  other  accord- 
ing as  they  are  directed  in  the  same  or  opposite  directions 
when  referred  to  the  vertex  of  the  angle. 

Take  the  particular  case  of  two  parallel  currents  z,  i'  , 
the  one  of  finite  length  /,  the  other  indefinite  ;  let  r  be 
their  distance  apart.  The  field  due  to  the  indefinite  current 
cy  in  every  point  of  the  other  conductor,  has  an  intensity 


The  electromagnetic  force  directed  along  r  will  give  for 
a  displacement  dr  of  the  current  i'  an  element  of  work 

dw  =  i'Mdr-,         (§  145) 
consequently  the  force  acting  between  the  currents  is 

-      dw        .     .         tir  , 
/  =  —  -  =  z3C/  =2—1. 
dr  r 

160.  Reaction  produced  in  a  Circuit  traversed  by  a 
Current.  —  The  lines  of  force  generated  by  a  current  across 
its  own  circuit  naturally  penetrate  by  the  negative  face,  and 
their  electromagnetic  action  upon  the  current  consequently 
tends  to  increase  the  surface  limited  by  the  conductors. 


ROTATION  AND   DISPLACEMENTS.  223 

If,  then,  a  circuit  of  irregular  form  is  traversed  by  a  cur- 
rent, the  conductors  tend  to  be  made  to  assume  a  circular 
shape,  which  corresponds  to  the  maximum  surface.  To 
render  this  effect  evident,  ,h6wever,  the  current  must  be 
sufficiently  strong  to  heat  the^res  enough  to  soften  them 
and  enable  them  to  obey  the  electromagnetic  forces. 


FIG.  79. 

The  reaction  of  a  current  upon  itself  can  be  shown  in  an- 
other way.  A  circuit  (Fig.  79)  is  completed  by  a  curved 
movable  conductor  floating  on  mercury  contained  in  two 
long  troughs.  When  a  current  is  set  up,  the  movable  con- 
ductor is  displaced  towards  the  right,  so  as  to  increase  the 
surface  of  the  circuit. 

161.  Explanation  of  Electromagnetic  Displacements 
based  on  the  Properties  of  Lines  of  Force. — Faraday  ac- 
counted for  magnetic  or  electromagnetic  actions  by  attribut- 
ing certain  properties  to  the  magnetic  lines  of  force.*  In- 
stead of  admitting,  as  did  the  physicists  who  preceded  him, 
that  currents  and  magnets  act  at  a  distance  without  inter- 
mediary, he  was  convinced  that  their  reactions  are  produced 
by  means  of  a  medium  in  a  state  of  tension  or  movement 
such  that  the  currents  and  magnets  thus  undergo  the  ob- 
served actions. 

As  we  have  seen  in  §  47,  Faraday  imagined  the  medium 
as  tense  along  the  lines  of  force,  for  which  he  substituted, 

*  Faraday,  Experimental  Researches. 


224  ELECTROMAGNETISM. 

in  thought,  elastic  threads  having  a  tendency  to  shorten, 
while  choosing  the  path  that  is  most  permeable  magnetically. 

The  curved  form  of  the  lines  of  force  is  explained  in  this 
hypothesis  by  the  mutual  repulsion  of  neighboring  lines  hav- 
ing the  same  direction.  On  the  other  hand,  if  the  neighbor- 
ing lines  have  opposite  directions,  they  attract  each  other  and 
tend  to  unite. 

These  practical  rules  suffice  to  predict  all  electromagnetic 
actions.  Given  a  system  of  magnets  and  currents,  it  is  al- 
ways possible  to  represent  mentally  the  distribution  of  the 
lines  of  force  due  to  each  of  the  elements  of  the  system. 
On  then  combining  these  lines  according  to  the  directions 
given  above,  we  obtain  the  form  of  the  resultant  field,  to 
which  we  need  only  apply  the  first  of  the  above  rules  in 
order  to  determine  the  direction  of  the  displacement  which 
will  occur. 

But  it  is  easier  to  find  the  resultant  field  directly  by  using 
the  method  of  magnetic  figures  (§47).  The  iron-filings  show 
the  distribution  of  lines  of  force,  which  take  the  form  of 
closed  curves  around  currents  and  of  interrupted  lines  in 
magnets.  By  their  tendency  to  shorten,  the  former  are 
urged  to  assume  a  circular  shape  as  having  a  minimum  per- 
imeter, and  the  latter  a  rectilinear  form.  By  means  of  these 
magnetic  figures  the  movements  of  the  system  can  be 
clearly  foreseen. 

As  an  example,  let  us  consider  a  small  straight  magnet 
capable  of  turning  between  the  limbs  of  a  horseshoe  mag- 
net (Fig.  80).  The  iron-filings  figure  shows  that  the  line's 
of  force  of  the  horseshoe  magnet  are  bent,  and  that  a  por- 
tion of  them  enter  the  small  magnet  by  its  s  end,  leaving  by 
its  n  end.  Their  tendency  to  shorten  causes  the  small  mag- 
net to  place  itself  in  the  direction  of  the  line  joining  the 
poles  of  the  other  magnet. 

The  same  thing  happens  if  we  replace  the  straight  mag- 


ROTATION  AND    DISPLACEMENTS. 


22$ 


net  by  a  circular  current  (Fig.  81).     The  lines  of  force  from 
the  horseshoe  magnet  will  traverse  the  current  through  its 


Vfe^^.-;    fe^ 


FIG.  80. 


FIG.  81. 


negative  face  and  force  it  to  assume  a  position  normal  to 
the  line  between  the  poles. 

In  the  case  of  Barlow's  wheel  (§  157),  the  lines  of  force 
from  the  magnet  are  shifted  by  those  of  the  current  travers- 
ing the  wheel.  The  result  is  a  field  whose  lines  embrace 
the  movable  disc  and  pull  it  in  one  direction  or  the  other, 
according  to  the  direction  of  the  current. 

So,  too,  two  parallel  currents  in  the  same  direction  com- 
bine their  fields  in  such  a  way  as  to  furnish  lines  which  em- 
brace both  conductors,  and  oblige  them  to  approach  each 
other  (Fig.  82). 


»N 


FIG.  82. 


FIG.  83. 


If  the  currents  are  in  opposite  directions,  the  circular  lines 
of    force    surrounding   each    conductor   assume   an    oblong 


226  ELECTROMA  ONE  TISM. 

shape  :  to  enable  them  to  return  to  their  circular  form,  it  is 
necessary  for  the  currents  to  move  apart. 


ELECTROMAGNETS. 

162.  Since  an  electric  current  can  produce  intense  mag- 
netic fields,  as  appears  from  the  investigation  of  straight 
and  annular  solenoids  (§  151),  it  enables  us  to  obtain  perma- 
nent or  temporary  magnets  of  considerable  magnetic  power. 
Thus  a  core  of  soft  iron,  surrounded  by  a  bobbin  of  insu- 
lated wire,  forms  a  temporary  magnet  when  a  current  trav- 
erses the  bobbin.  A  steel  core  after  the  passage  of  the 
current  preserves  a  permanent  magnetization  in  proportion 
to  the  degree  of  hardness  of  the  metal. 

The  system  composed  of  this  core  and  bobbin  is  called  an 
electromagnet.  The  poles  of  the  electromagnet  are  of  the  same 
sign  as  the  corresponding  ones  of  the  magnetizing  bobbin. 
If  two  straight  electromagnets,  parallel  and  placed  in  oppo- 


FIG.  84. 

site   directions,  be   connected  by  a  cross-piece   of  soft   iron 
called  the  yoke,  we  obtain  a  horseshoe  electromagnet. 

The  intensity  of  magnetization  of  the  core  depends  at 
each  point  on  the  magnetizing  force,  which  is  the  resultant 
of  the  action  of  the  bobbin  and  the  induced  magnetism 

(§  52)- 

It  follows  that  the  magnetism  of  the  core  can  only  be  cal- 
culated in  certain  simple  cases.  If,  for  example,  the  bobbin 
is  straight  and  indefinitely  long,  as  well  as  the  core,  the 
field  due  to  the  bobbin  is  constant  in  the  medial  region,  and 


ELECTROMA  GNE  TS.  22/ 

the  effect  of  the  induced  poles  towards  the  ends  of  the  core 
is  zero  in  this  region. 

If  we  denote  by  nl  the  number  of  turns  of  wire  per  unit 
length  along  the  axis  of  ttye*  bobbin,  by  i  the  current,  the 
field  due  to  the  current  is  exj^ressed  by  (§  151) 


OC  = 
The  magnetic  induction  in  the  core  is  (§  60) 


Let  5  be  the  section  of  the  bobbin,  S'  that  of  the  core; 
the  total  flux  across  the  bobbin  will  be  expressed  by 

$  =  47tnli(S  +  47T/C.S'). 


In  the  case  of  an  annular  core  (§  153)  we  have  an  identi- 
cal expression,  when  the  thickness  of  the  ring  is  small  rela- 
tively to  its  diameter. 

Towards  the  ends  of  a  straight  electromagnet  the  flux 
across  its  different  normal  sections  is  not  constant  ;  the  in- 
tensity of  magnetization  of  the  core  is  feebler  there  than  in 
the  middle  on  account  of  the  demagnetizing  action  of  the 
poles.  This  can  be  shown  by  the  figures  formed  by  iron- 
filings,  which  show  lines  of  force  emerging  laterally  from 
the  core,  beginning  at  a  certain  distance  from  the  extremi- 
ties. 

163.  Energy  expended  in  Electromagnets.  General 
Definition  of  the  Coefficient  of  Self-induction  of  a  Cir- 
cuit. —  We  have  seen  that  the  energy  of  a  current  due  to  its 
field  is  expressed  by 

^f,        (§144) 
Ls  denoting  the  flux  traversing  the  circuit   for  a  current 


228  ELECTROMA  GNE  TISM. 

equal  to   unity,  on  condition  that   the  permeability  of  the 
medium  is  constant. 

In  the  case  of  a  very  long  bobbin,  the  flux  across  one  of 
its  spirals  is  4rtnls.  If  there  are  n  spirals  and  if  we  neglect 
the  irregularity  of  the  field  towards  the  extremities,  we  have 


The  intrinsic  energy  is 


When   an   iron   core  is  introduced,  of  section  s,  the  flux 
corresponding  to  a  current  i  becomes  approximately 


and  the  energy 


The  difference 

27tnln(47tKs')?  = 

corresponds  to  the  work  of  magnetization  of  the  core. 

We  can  also  designate  by  the  name  of  coefficient  of  self- 
induction  of  the  electromagnet  the  relation  L/  =  4?tnln 
(s  +  4?rKs')  of  the  flux  to  the  current;  but  it  will  be  observed 
that  this  relation  is  no  longer  constant,  as  in  the  case  of  a 
simple  bobbin  :  it  contains  the  parimeter  /c,  which  is  vari- 
able with  the  current. 

In  the  case  where  a  section  of  the  bobbin  can  be  con- 
sidered as  equal  to  that  of  the  core,  we  have 

L  ',  =  47tnlnfjis. 

The  coefficient  of  self-induction  is  then  proportional  to 
the  permeability  of  the  core.  The  coefficient  of  self-indue- 


ELECTROMA  GNE  TS.  22Q 

tion  of  a  coreless  bobbin  is  invariable,  because  the  per- 
meability of  air  may  be  considered  as  constant  and  equal  to 
unity  (§  60). 

In  a  general  way  we  see  thaj:  the  coefficient  of  self-induc- 
tion of  a  circuit  is  the  ratio  o&<the  flux  of  magnetic  force 
across  it  to  the  current ;  we  cannot  define  the  coefficient  as 
equal  to  the  magnetic  flux  for  unit  current  unless  the  circuit 
is  remote  from  any  object  made  of  iron,  nickel,  or  cobalt. 

From  the  moment  when  the  core  acquires  its  normal 
magnetization,  which  depends  on  the  current  and  its  pre- 
vious magnetic  conditions  (§  57),  the  effect  of  the  current  is 
limited  to  maintaining  the  magnetization,  /.  e.,  the  molec- 
ular orientation  of  the  core,  and  to  heating  the  wire  in  the 
bobbin. 

It  is  to  be  noted  that  a  solenoid  of  given  volume  can  be 
composed  of  a  small  number  of  turns  of  coarse  wire,  or  a 
large  number  of  fine  wire.  If  we  neglect  the  space  taken 
up  by  the  insulation  on  the  wire  and  the  interstices  between 
the  coils,  we  find  that  the  electromagnetic  action  and  the 
heating  are  constant  as  long  as  the  current  remains  propor- 
tional to  the  cross-section  of  the  wire.  In  fact,  the  mag- 
netizing force  is  proportional  to  the  product  of  the  number 
of  turns  by  the  current ;  now  the  first  of  these  quantities  is 
in  inverse  ratio  to  the  cross-section  of  the  wire,  while  the 
second  is  proportional  to  it.  On  the  other  hand,  the  heat 
produced  in  one  second  is  measured  by  the  product  of  the 
square  of  the  current  into  the  resistance  of  the  solenoid, 
but  with  a  constant  volume  the  total  resistance  is  in  inverse 
ratio  to  the  square  of  the  cross-section  of  the  wire  ;  conse- 
quently the  product  remains  constant. 

We  arrive  at  the  same  result  by  considering  the  density  of 
the  current  in  the  solenoid,  that  is  to  say,  the  current  per 
unit  of  section  of  the  conductor.  The  magnetizing  force 
and  the  heating  are  respectively  proportional  to  the  first 


230  ELECTROMAGNETISM. 

and   second    powers  of   the  density :    these  quantities  are 
therefore  constant  as  long  as  the  density  is  invariable. 

As  the  heating  effect  increases  much  more  rapidly  than 
the  magnetic  effect,  it  will  be  evident  that  it  is  the  heating 
which  limits  the  results  that  can  be  obtained  with  a  sole- 
noid. 

164.  Magnetic  Circuit.  Magnetomotive  Force.  Mag- 
netic Resistance  or  Reluctance.— A  knowledge  of  the 
magnetic  induction  across  electromagnets  is  very  impor- 
tant as  regards  their  application  in  the  construction  of  ma- 
chines. 

To  determine  the  flux  of  magnetic  force,  it  must  be 
remembered  that  the  lines  of  force  generated  by  a  current 
are  continuous  curves  closed  upon  themselves  in  a  homo- 
geneous or  heterogeneous  medium  (§  151).  In  the  case  of 
an  annular  core,  completely  surrounded  by  a  solenoid,  the 
lines  are  concentrated  in  the  core  (§  153).  In  the  case  of  a 
straight  electromagnet  the  flux  of  force  traversing  the  sole- 
noid makes  its  exit  from  the  positive  region,  and  returns  by 
the  negative  region  after  diffusing  itself  through  the  sur- 
rounding space. 

The  law  of  continuity,  which  is  obeyed  by  the  magnetic 
flux  as  well  as  by  the  electric  flux  or  current,  leads  us  to 
consider  whether  the  two  categories  of  phenomena  are  not 
governed  by  the  same  laws.  We  have  seen  that  different 
substances  conduct  the  lines  of  magnetic  force  differently, 
and  we  have  given  the  name  permeability  to  the  coefficient 
which  characterizes  a  body  in  this  respect.  Hopkinson  and 
Kapp  have  compared  this  coefficient  to  the  coefficient  of 
conductivity  for  the  electric  flux.  They  have  thus  been  led 
to  apply  to  the  magnetic  circuit,  traversed  by  a  greater  or 
less  flux  according  to  its  permeability,  similar  relations  to 
those  which  govern  the  current  in  an  electric  circuit. 


ELECTROMA  GNE  TS.  2  3  1 

Take  the  simple  case  of  a  homogeneous  circuit  formed  of 
an  iron  ring  completely  surrounded  by  a  solenoid  traversed 
by  a  current.  The  magnetic  induction  is 


(B  =  /tfC,  where^JC  =  ^rrnj.         (§  153) 
The  total  flux  across  the  core,  of  section  s,  is 
$  =  ($>s  = 


Calling  /  the  length  of  the  circular  axis  of  the  ring,  n  the 
total  number  of  turns  in  the  solenoid, 


whence 

(i) 


This  form  is  analogous  to  Ohm's  law  ;  the  magnetic  flux 
is  proportional  to  the  expression  ^uni,  which  by  similitude 
is  called  magnetomotive  force*  and  inversely  proportional  to 

— ,  which  is  given  the  name  of  magnetic  resistance  of  the 
circuit,   on    account   of   its   likeness   to   electric    resistance 

(i  us). 

Heaviside,  in  consideration  of  the  fact  that  there  is  in 
magnetism  nothing  analogous  to  the  Joule  effect,  which  is 
work  accomplished  by  the  electric  current  to  overcome  the 
electric  resistance,  has  substituted  for  the  expression  "mag- 
netic resistance  "  that  of  reluctance. 

We  can  get  equation  (i)  by  a  direct  proceeding.  We 
have  seen  (§  141)  that  the  work  performed  by  a  unit  pole 

*  Bosanquet,  Philosophical  Magazine,  1883,  Vol.  XV, -p.  205. 


232  ELECTROMA  ONE  TISM. 


in  moving  across  a  circuit  is  equal  to  ^ni  times  the  number 
of  revolutions  made,  ^ni  representing  the  difference  of  mag- 
netic potential  between  the  two  faces  of  the  circuit.  Now 
if  the  unit  pole  moves  along  the  internal  axis  of  the  ring,  it 
traverses  the  current  n  times  in  one  revolution.  The  work 
accomplished,  ^.nni  is  also  expressed  by  the  product  of  the 
mean  intensity  3C  into  the  path  /  traversed  by  the  unit  pole, 
whence 


but 

whence 

#  = 


The  magnetomotive  force  ^nni  is  therefore  measured  by 
the  sum  of  the  differences  of  magnetic  potential  produced 
in  the  solenoid. 

If  the  core  were  composed  of  various  segments  having 
lengths  /,  /',  I",  sections  s,  s',  s",  and  permeabilities  //,  //,  /*", 
we  would  have 


There  is,  however,  an  essential  distinction  to  be  made  be- 
tween the  electric  and  magnetic  circuits.  The  resistance  of 
the  former  is  independent  of  the  current,  while  that  of  the 
latter  is  a  function  of  the  permeability,  which  depends  not 
only  on  the  actual  flux,  but  also  on  the  preceding  fluxes 
(§  57)-  On  tne  °ther  hand,  the  flux  and  the  quantity  of 
magnetism  are  not  connected  by  a  law  similar  to  that  which 
connects  the  quantity  of  electricity  to  the  current  and  the 


ELECTROMAGNETS.  233 

time.  Finally,  in  consequence  of  the  residual  magnetism,  a 
magnetic  flux  can  exist-in  a  circuit  without  any  magnetomo- 
tive force,  or  even  a  flux  in  a  direction  opposed  to  the  mag- 
netomotive force.  We  must  therefore  be  careful  not  to  con. 
elude  an  analogy  of  fact  from  a^i  analogy  of  form,  and  only 
look  on  the  extension^of  Ohm's  law  to  the  magnetic  circuit 
as  an  artifice  to  facilitate  the  investigation  of  the  question 
as  well  as  the  calculations  connected  with  it. 

Following  this  order  of  ideas,  we  can  push  the  compari- 
son still  further  and  treat  cases  of  complex  magnetic  cir- 
cuits by  Kirchhoffs  laws. 


FIG.  85. 

Let  us  take  the  case  of  a  magnetic  circuit,  placed  in  a 
medium  supposed  to  be  impermeable  to  the  lines  of  force, 
and  divided  into  two  portions.  The  flux  $  generated  by  a 
solenoid  of  n  turns  traversed  by  a  current  i,  divides  itself 
into  two  derived  fluxes,  $'  and  3>". 

We  shall  have 

$=&+  <£". 

Let  /  be  the  length  of  the  common  branch,  s  its  section, 
and  /i  its  permeability. 

Let  I' ,  /,  //  ;  I" ,  s" ,  //'  be  the  corresponding  elements  for 
the  derived  branches. 


2  34  ELECTROMA  GNE  TISM. 

Kirchhoff's  second  law  will  give  the  equations 


The  case  that  we  have  just  considered  is  imaginary,  for 
there  is  no  medium  in  existence  which  is  impermeable  to 
lines  of  force.  In  the  phenomena  of  the  electric  current  we 
practically  take  air  at  the  ordinary  pressure  as  a  perfect  in- 
sulator for  the  electric  flux ;  but  the  same  does  not  hold 
for  magnetic  phenomena.  If  the  permeability  of  air  and 
other  gases  is  negligible  compared  with  that  of  iron  when 
traversed  by  a  mean  magnetic  induction,  it  becomes  de- 
cidedly comparable  with  it  for  intense  magnetic  fluxes.  It 
follows  that  in  the  case  of  Fig.  85,  for  example,  a  part  of 
the  flux  generated  by  the  solenoid  will  go  off  into  the  sur- 
rounding medium.  The  ratio  of  the  flux  into  the  air  to  the 
flux  traversing  the  iron  rings  will  increase  with  the  current 
in  the  solenoid  in  consequence  of  the  gradual  weakening  of 
the  permeability  of  iron  and  the  constancy  of  that  of  air. 

The  case  is  not  without  analogy  with  that  of  an  electric 
circuit  placed  in  the  midst  of  a  liquid  having  a  certain  rela- 
tive conductivity.  A  part  of  the  current  furnished  by  the 
electric  generator  will  pass  through  the  liquid  without  fol- 
lowing the  line  of  the  metallic  circuit.  We  shall  see,  when 
studying  dynamos,  how  the  derived  flow  of  the  magnetic 
flux  in  the  surrounding  medium  can  be  experimentally  es- 
timated. 

165.   Forms  and  Construction  of  Electromagnets. — 

When  it  is  desired  to  utilize  the  portative  power  of  an  elec- 
tromagnet, the  horseshoe  form  is  used,  consisting  of  two 
straight  cores  wound  with  wire  and  connected  at  the  base 


ELECTROMA  GNE  TS.  235 

by  an  iron  yoke.  The  free  poles  attract  the  armature  (or 
keeper],  which,  with  the  air-gap,  completes  the  magnetic 
circuit. 

The  attractive  force  is  proportional  to  the  cross-section 
and  the  square  of  the  magnetic  induction.  We  have  seen 
(§  56)  that  the  portative'  power  of  an  electromagnet  per  unit 

/T>2 

section  is   2n??  +  OC3,  which   expression  becomes  -  -  if  we 

O7t 

'TO2 

add   the  very  small  term  —  .     It  is  therefore  advantageous 

Q7T 

to  make  electromagnets  of  short  and  massive  pieces. 

When  we  wish  to  calculate  an  electromagnet  to  obtain  a 
given  portative  power 


we  take  a  magnetic  induction  (B,  which  can  be  obtained 
without  too  great  expenditure  of  electric  energy.  The 
magnetism-curves  (§  57)  show  that  beyond  (E  =  16000  the 
increase  in  induction  is  very  feeble  relatively  to  the  ex- 
penditure :  consequently  we  choose  a  section  s  about 


16000 

Next  we  seek  the  dimensions  of  the  bobbins  capable  of  pro- 
ducing  the  flux  <$>s  ;  we  take  a  given  length,  /,  for  the  cores, 
and,  after  having  joined  them  by  a  yoke  and  a  keeper  of 
section  s,  we  calculate  the  magnetomotive  force  qnni  that 
the  coils  must  develop,  by  the  formula 

=  ($>s  X  — 


An  example  of  such  a  calculation  will  be  seen  further  on. 
Instead  of  using  the  horseshoe  form  in  order  to  obtain  a 


ELECTROMA  GNE  TISM. 


closed  circuit,  the  helix  of  a  straight  electromagnet  may  be 
enclosed  by  an  iron  sheath  joined  to  the  core  at  one  end  by 
the  yoke,  and  at  the  other  by  the  keeper,  Fig.  86.  This 


FIG.  86. 

form  is  very  compact,  but  it  does  not  lend  itself  readily  to 
the  employment  of  strong  currents,  for  the  radiation  of  the 
heat  produced  in  the  helix  by  the  Joule  effect  does  not  take 
place  so  well  as  in  the  other  form. 

When  the  keeper  is  moved  away  from  the  poles,  the  at- 
tractive force  diminishes  very  rapidly.  In  fact  the  attrac- 
tion is  inversely  proportional  to  the  square  of  the  distance 
between  the  poles  (of  the  keeper  and  magnet)  which  are 
opposite  each  other,  and  the  poles  themselves  decrease  with 
the  magnetic  flux,  which  grows  very  suddenly  less  on  ac- 
count of  the  feeble  permeability  of  the  air.  It  follows  that  at 
quite  a  short  distance  the  attraction  exercised  by  the  mag- 
net-poles on  the  keeper  becomes  negligible ;  we  have,  conse- 
quently, to  limit  the  amount  of  play  of  this  movable  part. 

In  order  to  diminish  the  weakening  of  the  flux,  caused 
by  the  large  reluctance  of  the  air-gap,  we  can  increase  the 
magnetomotive  force  by  increasing  the  number  of  turns  in 
the  helix.  This  leads  to  the  lengthening  of  the  cores  ;  this 
lengthening  produces  in  itself  an  increase  in  the  magnetic 
resistance,  but  in  view  of  the  high  value  of  the  permeability 
of  iron  compared  with  that  of  air,  the  total  resistance  of  the 
circuit  is  not  notably  increased.  This  means  is  made  use 
of  in  telegraphic  electromagnets. 

In  many  applications  it  is  necessary  to  increase  the  play 


ELECTROMA  GNE  TS.  % 

of  the  keeper ;  this  is  managed  very  simply  by  giving  it,  in- 
stead of  a  movement  perpendicular  to  the  line  of  the  poles, 
an  oblique  movement,  which  augments  its  play  without 
altering  the  work  done  in  displacement.  One  solution  con- 
sists in  furnishing  the  keeper  w^h  conical  projections  which 
penetrate  into  conical  cavities  cut  in  the  cores. 

If  it  is  desired  to  obtain  considerable  displacements,  with 
a  small  variation  in  the  attractive  force,  we  must  make  use 
of  the  suction-effect  of  a  long  solenoid  upon  a  cylindrical 
core  placed  at  the  open  end  of  the  solenoid.  The  core  tends 
to  place  itself  in  such  a  position  that  the  flux  traversing  it  is 
a  maximum  (§§  65,  151). 

Before  analyzing  the  effect  of  a  long  bobbin  or  solenoid, 
placed  vertically,  upon  a  core  acting  as  a  plunger,  it  should 
be  recalled  that  the  field  is  sensibly  uniform  in  the  middle 
region  of  the  solenoid,  and  that  its  intensity  decreases  rap- 
idly towards  the  ends  (§  151).  A  pole  of  invariable  strength 
is  therefore  urged  through  the  solenoid  by  a  force  which  in- 
creases from  the  point  of  entry,  becomes  constant  in  the 
middle,  and  decreases  towards  the  lower  end.  The  force  is 
in  each  point  equal  to  the  intensity  of  the  field  multiplied 
by  the  strength  of  the  given  pole. 

When  a  soft  iron  core  is  presented  at  the  opening  of  a 
solenoid,  the  phenomenon  is  more  complex.  The  attractive 
effort  depends  on  the  magnetization  induced  in  the  core, 
which  itself  varies  with  the  intensity  of  the  field.  To  sim- 
plify the  matter,  let  us  consider  a  cylindrical  core  whose 
length  is  very  great  compared  to  that  of  the  solenoid,  so 
that  we  have  only  to  consider  the  action  exercised  upon  the 
pole  induced  in  its  lower  end.  The  strength  of  this  pole  in- 
creases as  the  core  descends,  for  the  total  flux  produced  by 
the  solenoid  increases  in  consequence  of  the  gradual  dimi- 
nution of  the  resistance  of  the  magnetic  circuit  due  to  the 
insertion  of  the  iron  core.  When,  however,  its  lower  end 


238  ELECTROMAGNETISM. 

reaches  the  bottom  of  the  solenoid  the  force  tends  to  de- 
crease, for  the  induced  pole  reaches  a  region  of  the  field 
where  the  intensity  is  diminishing. 

If  the  length  of  the  core  equals  that  of  the  solenoid,  we 
must  take  into  consideration  the  antagonistic  action  exer- 
cised on  the  pole  induced  in  the  upper  end  of  the  core. 
Experiment  shows  that  in  this  case  the  resultant  effort  is 
maximum  when  this  latter  reaches  the  middle  of  the  sole- 
noid ;  it  then  decreases  and  becomes  zero  when  the  core 
is  in  a  symmetrical  position  with  regard  to  the  solenoid,  this 
being  the  position  in  which  the  flux  produced  by  the  cur- 
rent and  traversing  the  iron  is  a  maximum. 

In  order  to  regulate  the  attractive  effort  exercised  on  the 
core,  when  this  latter  is  subject  to  considerable  displace- 
ments, it  is  given  the  form  of  a  very  long  cone  with  the  apex 
pointing  downwards.  In  this  case  the  total  flux  increases 
as  the  core  enters  into  the  solenoid,  even  when  the  apex  of 
the  cone  has  passed  the  lower  opening. 

Solenoids  develop  a  very  much  lower  attractive  effort  on 
their  cores  than  that  exercised  by  electromagnets  of  horse- 
shoe shape  on  their  keepers,  for  in  the  latter  case  the  mag- 
netic circuit  has  much  less  reluctance. 

To  increase  the  suction-effect  we  can  put  an  iron  sheath- 
ing on  the  solenoid,  which  presents  a  very  permeable  path 
for  the  lines  of  force.  A  solenoid  armored  in  this  way  gives 
a  very  intense,  uniform  field  in  its  interior,  but  its  external 
effects  are  negligible :  the  core  is  not  attracted,  in  this  case, 
until  it  is  introduced  into  one  of  the  openings  in  the  sheath- 
ing. 

In  order  to  reduce  as  much  as  possible  the  perimeter  of 
the  wire  in  an  electromagnet,  the  core  must  have  a  circular 
section  ;  sometimes,  however,  a  square  or  rectangular  sec- 
tion is  chosen  to  gain  compactness.  Such  a  shape  increases 
not  only  the  length  and  consequently  the  resistance  of  the 


ELECTROMA  GNE  TS.  239 

magnetizing  wire,  but  also  the  losses  of  flux  which  take 
place  between  the  lateral  faces  of  the  adjoining  cores,  par- 
ticularly in  horseshoe  electromagnets,  by  bringing  the  cores 
closer  together  and  increasing  theJr  surface. 

As  far  as  possible  sharp  edges  on  the  polar  surfaces  must 
be  avoided,  since  the  liftes  of  force  have  a  tendency  to 
crowd  together  at  the  edges  and  escape  by  them  into  the 
surrounding  air.  This  effect  is  not  without  analogy  with 
the  effect  of  points  in  the  case  of  electrified  bodies. 

The  wire  used  in  electromagnets  is  generally  of  the  purest 
possible  copper  so  as  to  reduce  the  heating.  If  the  cross- 
section  of  the  conductor  has  to  be  large,  it  is  made  more 
manageable  by  using  cords  of  twisted  copper  strands  or 
bundles  of  copper  strips.  These  latter  have  the  advantage 
of  diminishing  the  interstices  between  the  consecutive  turns. 
For  mean-tension  currents  the  wire  is  covered  with  two  or 
three  layers  of  cotton  impregnated  with  varnish  or  shellac. 
If  the  coils  are  very  small,  the  wire  is  generally  insulated 
with  silk.  For  high-tension  currents  the  successive  layers 
of  wire  must  be  separated  by  vulcanized  fibre,  cotton  tape 
impregnated  with  shellac,  Wellesden  paper,  mica,  or  ebon- 
ite. In  special  cases  the  windings  are  even  separated  into 
parts  by  ebonite  partitions  perpendicular  to  the  axis  of  the 
core,  the  spaces  between  the  partitions  being  filled  with  flat 
bobbins  connected  in  series.  In  this  manner  conductors  at 
very  different  potentials  are  kept  apart  and  the  danger  of 
disruptive  sparks  avoided. 

If  electromagnets  are  exposed  to  excessive  heating,  the 
wire  is  insulated  with  asbestos  or  mica. 

The  conductor  joining  the  inner  layer  of  a  coil  with  the 
outer  <  layer  should  be  made  extra  strong,  for,  if  it  breaks, 
the  coil  has  to  be  unwound  in  order  to  repair  it. 

The  shape  given  to  the  keeper  of  an  electromagnet  has  a 
marked  influence  on  the  magnet's  portative  power,  which  is 


24O  ELECTROMA  GNE  TfSM. 

(B1 
expressed  by  —  s  (§  165).      If  the  polar  surface  is  dimin- 

O7f 

ished,  the  induction  (B  is  increased,  for  the  lines  of  force, 
having  a  tendency  to  make  their  way  across  the  iron  rather 
than  across  the  surrounding  air,  will  go  more  and  more  to- 
wards this  surface  in  proportion  as  it  grows  smaller  near  the 
keeper.  It  follows  that  the  product  ($>*s  will  be  a  maximum 
for  a  value  of  s,  generally  much  smaller  than  the  section  of 
the  cores.  It  is  for  this  reason  that  keepers  are  frequently 
given  a  convex  form  in  the  part  adjoining  the  poles. 

166.  Magnetization  of  a  Conductor. — When  a  perma- 
nent electric  current  traverses  an  iron  or  steel  wire,  the  mole- 
cules situated  on  the  surface  of  the  wire  align  themselves 
circularly  along  the   lines  of   force   created  by  the   interior 
electric  flux,  forming  closed  magnetic  filaments  without  ex- 
ternal action.     To  exhibit  this  transverse  magnetization,  we 
need  only  cut  a  longitudinal  groove  in  the  wire ;  its  edges 
will  exhibit  opposite  polarities. 

167.  Modifications  in  the  Properties  of  Bodies  in  a 
Magnetic  Field. — A  magnetic  field  causes  perturbations  in 
the  propagation  of  light  in  bodies  placed  in  the  field.     This 
discovery,  due  to  Faraday,  is  proven  by  the  aid  of  polarized 
light,  whose  plane   of  polarization  is  altered  when   it  trav- 
erses, in  the  direction  of  the  lines  of  force,  a  solid,  liquid,  or 
gaseous  body  placed  in  a  magnetic  field. 

Faraday  used,  in  his  demonstration,  an  electromagnet 
whose  field-cores  are  hollowed  out  along  their  axis  and 
joined  by  right-angled  iron  supports  and  an  iron  base. 

The  substance  to  be  tried  is  placed  between  the  poles  of 
the  magnet.  The  field-cores  are  traversed  by  a  ray  of  light 
polarized  by  passing  through  a  Nicol's  prism  ;  on  leaving 
the  field-cores  the  ray  is  extinguished  by  an  analyzer.  When 
a  current  is  sent  through  the  field-coils  the  ray  reappears : 


ELECTROMAGNETS.  24! 

it  can  be  re-extinguished  by  giving  the  analyzer  a  rotation 
which  measures  the  angle  by  which  the  plane  of  polariza- 
tion has  turned. 

Verdet  has  shown  that  this  angle  is  proportional  to  the 
difference  of  the  magnetic  potentials  at  the  extreme  points 
of  the  path  of  the  ray  trlrough  the  substance  affected.  The 
direction  of  rotation,  however,  is  independent  of  the  direc- 
tion of  the  ray  of  light,  but  is  different  in  magnetic  and  dia- 
magnetic  bodies. 

This  experiment  can  also  be  made  by  the  aid  of  a  sole- 
noid traversed  by  a  current  and  having  placed  along  its  axis 
a  tube,  closed  by  glass  ends,  in  which  the  liquid  to  be  ex- 
perimented on  has  been  poured.  The  polarizer  is  placed  at 
one  of  the  ends  of  the  tube  and  the  analyzer  at  the  other. 

If  the  length  of  the  tube  greatly  exceeds  that  of  the  sole- 
noid, the  magnetic  potential  is  practically  zero  towards  its 
extremities  (§  151);  the  difference  of  magnetic  potential  is 
equal  to  the  work  accomplished  by  unit  pole  in  moving  from 
one  of  these  extremities  to  the  other,  or  ^nni',  n  denoting 
the  total  number  of  turns,  and  i  the  current. 

a  being  a  constant,  called  Verdet's  constant,  for  a  given 
body  at  a  definite  temperature,  the  rotation  of  the  plane 
of  polarization  is 

0  = 


This  relation  gives  the  means  of  measuring  the  current  as 
a  function  of  the  deviation  of  the  polarized  ray. 

168.  Hall  Effect.  —  Hall  has  discribed  a  phenomenon 
which,  like  the  preceding,  probably  owes  its  origin  to  a 
physical  modification  in  the  bodies  placed  in  a  field. 

If  we  connect  with  the  poles  of  a  battery  the  points  a  and 
b  (Fig.  87)  of  a  thin  conductive  plate  of  circular  form,  we 
can  trace  equipotential  lines  and  lines  of  electric  flux  dis- 
tributed as  shown  in  the  figure.  In  order  to  obtain  one  of 


242 


ELECTROMA  GNE  TISM. 


the  equipotential  lines,  we  need  only  affix  at  one  point  of 
the  plate  the  extremity  of  a  wire  connected  with  a  galva- 
nometer, and  move  about  over  the  plate,  another  wire  con- 


nected  to  the  second  terminal  of  the  same  instrument. 
Each  time  that  we  meet  with  a  point  at  the  same  potential 
as  the  first,  the  galvanometer  will  show  no  deviation. 

The  equipotential  lines,  marked  by  dots,  are  symmetrical 
to  the  diameter  cd,  whose  extremities  are  at  the  same  po- 
tential. 

Suppose  now  that  the  disc  be  placed  between  the  poles 
of  an  electromagnet  so  that  the  lines  of  force  traverse  it 
perpendicularly. 

The  points  c  and  d  immediately  cease  to  be  at  the  same 
potential,  and  a  new  distribution  of  the  lines  of  flux  is  pro- 
duced as  shown  in  Fig.  88. 


FIG.  88. 


At  first  sight  it  seems  as  if  the  displacement  of  the  lines 
of  flux  ought  to  be  attributed  to  the  direct  action,  governed 


ELECTROMAGNETS.  243 

by  Ampere's  rule,  of  the  magnetic  force  on  the  electric  cur- 
rent. 

But  experiment  shows  that  the  deviations  are  not  in  the 
same  direction  in  different  bodies.  For  example,  the  defor- 
mation of  the  lines  of  flux  follow^  Ampere's  rule  in  iron  and 
zinc  ;  it  is  the  opposite  In  bismuth  and  nickel. 

At  the  same  time  with  the  above-mentioned  displace- 
ment, there  is  observed  an  apparent  increase  in  the  electric 
resistance  of  the  conductor,  attributable  to  the  fact  that  the 
mean  lines  of  flux  are  lengthened  by  the  twisting.  This  in- 
crease of  resistance,  which  is  more  manifest  in  conductors  of 
lengthened  form  placed  perpendicularly  to  the  lines  of  force, 
can  be  used,  as  M.  Leduc  has  shown,  to  measure  the  inten- 
sity of  a  magnetic  field. 


UNITS   AND   DIMENSIONS* 

GENERAL  THEORY. 

169.  Units. — In  every  quantitative  statement  of  the  value 
of  a  physical  quantity  two  factors  are  involved — the  unit  of 
the  same  concrete  kind  as  the  quantity,  and  a  numerical  fac- 
tor. The  numerical  factor,  called  the  "  numeric  "  for  short, 
is  the  ratio  of  the  concrete  quantity  to  its  unit ;  or,  it  is  the 
number  expressing  how  many  times  the  unit  is  contained  in 
the  concrete  quantity.  For  instance,  if  /  is  a  definite  length 
and  L  the  unit  length,  then  l/L  is  the  numerical  value,  or 
numeric,  of  this  length  /. 

It  is  clear  that  when  the  unit  chosen  as  a  measure  of 
length  is  changed,  the  numeric  of  any  concrete  length  will 
change  inversely ;  any  length  contains  twelve  times  as 
many  inches  as  feet,  the  unit  being  diminished  from  feet  to 
inches  in  the  ratio  of  I  :  12,  the  numeric  increases  in  the  ra- 
tio of  12:  i. 

A  unit,  such  as  that  of  length,  which  involves  in  its  defi- 
nition no  reference  to  other  units,  is  called  a  fundamental 
unit.  From  the  fundamental  unit  others  flow,  dependent 
upon  the  fundamental  units ;  these  are  called  derived  units. 

Such  derived  units  are  the  unit  of  area, — the  area  of  a 
square  whose  side  is  the  unit  length ;  the  unit  of  volume, 
— the  volume  of  a  cube  whose  edge  is  the  unit  length  ; 

*  By  Gary  T.  Hutchinson,  Ph.D. 

244 


GENERAL    7'HEORY.  245 

the  unit  of  velocity, — the  velocity  with  which  unit  length 
will  be  traversed  in  unit- time:  these  are  all  simple  derived 
units.  Others  are  more  involved,  requiring  several  steps  in 
their  reference  to  the  fundamental  units.  For  instance,  the 
unit  of  acceleration  is  defmed^to  be  the  acceleration  with 
which  the  velocity  is  increased  by  unity  in  unit  time ;  this 
involves  a  reference  to  time  and  velocity  which  in  turn  im- 
plies length  and  time. 

The  choice  of  fundamental  units  is  to  a  great  extent  ar- 
bitrary. Rational  systems  of  units  can  be  built  up,  start- 
ing from  many  different  sets  of  fundamental  units ;  the 
units  of  length,  mass,  and  time  are,  however,  almost  uni- 
versally adopted  as  fundamental.  Practically  all  physical 
quantities  can  be  expressed  in  terms  of  length,  mass,  and 
time. 

170.  Dimensions. — A  change  in  the  fundamental  units 
will  obviously  change  a  derived  unit.  The  amount  by  which 
a  derived  unit  is  affected  by  given  changes  in  the  fun.da- 
mental  units  is  determined  by  what  are  called  the  "dimen- 
sions," or  the  dimensional  equation  of  the  derived  unit. 
These  dimensions  express  the  manner  in  which  the  funda- 
mental units  enter  into  the  derived  unit. 

A  simple  illustration  of  the  method  of  deriving  dimen- 
sions is  that  of  velocity.  Let  v,  /,  and  /  denote  respectively 
a  concrete  velocity,  length,  and  time,  connected  by  the  rela- 
tion that  the  numerical  value  of  the  velocity,  or  its  numeric, 
is  equal  to  the  numerical  value  of  the  length  divided  by  that 
of  the  time ;  that  is,  that  the  given  length  /  would  be  de- 
scribed in  the  given  time  t  by  a  body  moving  with  the 
given  velocity  v.  Let  V,  L,  T  be  the  units  of  velocity, 
length,  and  time,  respectively;  then  the  numerics  of  ^,  /,  and 
/  are  v/V,  l/L,  t/T,  and  by  assumption, 

v/r=i/L+ 


246  UNITS  AND   DIMENSIONS. 

or 


This  equation  shows  that  the  numeric  of  the  velocity  v 
varies  inversely  as  the  unit  of  length  Z,  and  directly  as  the 
unit  of  time  T.  It  also  shows  that  the  unit  velocity  V 
varies  directly  as  the  unit  length  Z,  and  inversely  as  the 
unit  time  T,  since  the  concrete  quantities  v,  /,  and  /  are 
not  affected  by  changes  in  the  units  F,  Z,  7". 

The  equation 


is  known  as  the  dimensional  equation  of  F,  and  LT~l  are 
the  dimensions  of  V. 

It  is  clear  that  these  dimensions  are  dependent  entirely 
upon  the  fundamental  units  adopted  and  upon  the  defini- 
tion given  to  velocity  ;  also,  that  they  have  no  connection 
with  the  real  physical  nature  of  the  quantities  to  which  they 
refer.  As  an  illustration,  suppose  force,  mass,  and  time 
were  chosen  fundamental  units  ;  then  since 

force  =  mass  X  acceleration, 
or  f=mXa=mX  v/tt 

v=/X  t/m, 
and 


an  entirely  different  set  of  dimensions. 

In  the  same  manner  the  following  dimensions  are  de- 
duced from  the  ordinary  definitions  of  the  quantities  in- 
volved. 


DIMENSIONS   OF  SOME   PHYSICAL    QUANTITIES.  247 


DIMENSIONS   OF   SOME   PHYSICAL   QUANTITIES. 


Physical  Quantity.  Symbol. 

Length /' 

Mass r. . .    m 

Time / 

Area s 

Volume   V 

Velocity   v 

Angular  velocity GO 

Density 

Force f 

Work   W 

Power P 

Pressure / 

Moment  of  inertia. ,  K 


Defining 
Equation. 


Dimensions 

L 

M 
T 


GO  =  V/l 

=  m/T 
f  —  m  x  a 

w=/xt 

P=  W/t 


LMT-* 

DMT-* 

UMT-* 


DM 


The  other  mechanical  and  dynamical  quantities  can  read- 
ily be  deduced  from  these  in  the  same  manner. 

171.  C.  G.  S.  System  of  Units. — The  system  of  units 
adopted  by  the  entire  scientific  world  is  that  based  upon  the 
centimetre  as  the  unit  length,  the  gramme  as  the  unit  mass, 
and  the  second  as  the  unit  time. 

These  units  were  chosen  after  long  consideration  because: 
— they  admit  of  accurate  comparison  with  quantities  of  their 
own  kind,  the  units  can  be  determined  easily  and  accurately 
and  standardized  at  all  times  and  places,  and  they  lead  to 
simple  definitions  of  the  most  important  physical  quantities, 
with  their  relations  to  one  another. 

In  all  that  follows  L  will  denote  unit  length  or  one  centi- 
metre, J/the  unit  mass  or  one  gramme,  and  T  the  unit  time 
or  one  second. 

The   C.   G.  S.  system  of  units  applies  to    mechanical  as 


248  UNITS  AND    DIMENSIONS. 

well  as  electrical    quantities,   although    it    has    been    given 
prominence  mainly  through  its  electrical  applications. 


ELECTRICAL  AND   MAGNETIC  UNITS. 

172.  General  Considerations. — A  consistent  system  of 
electrical  and  magnetic  units  can  be  devised  with  any  one 
of  the  equations  defining  an  electric  or  magnetic  relation  as 
a  starting-point.  Each  system  so  formed  would  have  differ- 
ent dimensions  for  the  same  concrete  quantity,  and  these 
dimensions,  as  explained  above,  would  serve  to  show  the 
changes  in  the  various  units  when  the  fundamental  units 
were  changed,  and  consequently  the  ratio  of  the  various 
units  in  the  different  systems. 

Two  systems  have  been  built  up  in  this  way,  and  are  in 
use  to-day :  the  first,  the  Electrostatic  System  (E.  S.),  is 
based  on  the  definition  of  unit  quantity  of  electricity ;  the 
second,  the  Electromagnetic  System  (E.  M.),  is  based  on 
the  definition  of  unit  magnetic  pole.  In  order  that  mag- 
netic quantities  may  be  expressed  in  terms  of  the  electro- 
static units,  and  electric  quantities  in  terms  of  electromag- 
netic units,  some  connecting  link  between  the  two  sets  of 
phenomena  must  be  used  :  this  link  is  the  relation  devel- 
oped in  §  136,  that  the  magnetic  force  due  to  an  unlimited 
rectilinear  current  i  at  a  point  at  a  distance  /,  is  proportional 
to  the  current  i,  and  inversely  proportional  to  the  distance 
/.  That  is, 

Magnetic  force  =  a  X  i/l. 

If  3C  is  the  unit  magnetic  force  and  /the  unit  current, 

OC  =  /£-' (A) 

There  is  only  one  known  relation  connecting  electric  cur- 
rent  and  magnetic  force  ;  equation  (A)  is  one  expression 


ELECTRICAL  AND    MAGNETIC   UNITS.  249 

of  this  relation.  Another  common  form  of  expressing  the 
same  relation  is, — the  work  done  by  unit  pole  in  passing 
around  any  closed  curve  that  threads  the  electric  circuit 
equals  47*7.  These  two  relations,  and  others  of  the  same 
kind,  are  interchangeable. 

Coulomb's  experiments  connect  mechanical  force  with 
electric  quantity  ;  as  ordinarily  stated, 

f=q-/r. 

This  statement  assumes  the  action  to  take  place  in  air  (or 
vacua),  and  does  not  bring  the  medium  into  evidence;  for 
greater  generality  the  expression  should  be 

f=f/(l'K),     .     .     ..-.     ,    .  (B) 

where  K  is  the  specific  inductive  capacity  or  permittivity 
of  the  medium. 

Similarly,  for  magnetic  quantities, 

/=*•/(/';•),     .     .     .     .     .     .    .  (C) 

where  JA  is  the  permeability  of  the  medium. 

173.  Systems  in  Terms  of  K  and  /*.— Starting  with  equa- 
tions (A)  and  (B),  a  system  of  units  can  be  developed  in 
which  all  the  electric  and  magnetic  quantities  are  expressed 
in  terms  of  LMT  and  K.  Similarly,  starting  with  (A)  and 
(C),  a  system  can  be  developed  in  which  all  the  quantities 
are  expressed  in  terms  of  LMT  and  ju. 

These  two  systems  are  the  only  ones  in  use :  the  first,  the 
system  in  terms  of  K,  leads  directly  to  the  Electrostatic 
System  ;  the  second,  the  system  in  terms  of  yu,  leads  to  the 
Electromagnetic  System.  The  two  are  herein  developed 
side  by  side ;  the  quantities  are  chosen  in  pairs  so  as  to 
show,  as  far  as  possible,  the  similarity  of  each  in  the  AT-sys- 


250  UNITS  AND   DIMENSIONS. 

tern  to  its  fellow  in  the  yw-system.  Units  in  terms  of  K  will 
be  distinguished  by  a  line  over  the  symbol. 

Electric  Quantity  (q)  : 

In  terms  of  K.  —  Unit  quantity  is  defined  by  equation  (B); 
that  is,  the  mechanical  force  /acting  between  two  equal 
quantities  q  at  distance  /  is 


=  FDK 


~Q  = 
Electric  Current  (Y)  : 


Electric  Current  (Y)  : 

In  terms  of  K.  —  Current  is   defined   as  the   quantity  of 
electricity  carried  across  the  section  of  a  conductor  in  unit 


time  ;    or, 

i  =  q/t, 

~T  


Strength  of  Magnetic  Pole  (m)  : 

In  terms  of  ^.  —  Pole  strength  is  defined  by  equation  (C), 
similar  to  equation  (B)  for  quantity.     Therefore, 


m  = 


Magnetic  Force  (3C)  : 

In  terms  of    //.  —  Magnetic   force  is  defined  as  the  force 
acting  on  unit  pole  ;  therefore, 

X  =  F/m 


ELECTRICAL   AND   MAGNETIC   UNITS.  251 

Electric  Current  (i)  : 

In  terms  of  /*.  —  Equation  (A),  the  linking  equation,  gives 
the  magnetic  force  due  to  unit  current  i  at  distance  /as 

OC  =  a  X  *//, 

. 


Electric  Quantity  (q)  : 
In  terms  of  //.  — 

q  =  i  x  /,  as  above 


Magnetic  Force  (X)  : 
In  terms  of  AT.  — 


Strength  of  Magnetic  Pole  (m)  : 
In  terms  of  K.  — 

f  —  JC;«,  as  above  ; 


All  the  electric  magnitudes  depend  on  q,  from  which 
they  are  easily  derived,  following  the  usual  definitions; 
hence,  having  ^expressed  both  in  terms  of  A^  and  of  /*,  the 
expression  for  all  the  electric  magnitudes  can  be  deduced  in 


252  UNITS  AND   DIMENSIONS. 

terms,  both  of  K  and  of  /*.  Similarly,  having  pole  strength 
m,  expressed  both  in  terms  of  K  and  of  /*,  the  expression 
for  all  the  other  magnetic  magnitudes  can  be  deduced  both 
in  terms  of  K  and  /*. 

The  accompanying  table  gives  the  symbol,  the  defining 
equation,  and  the  dimensions,  both  in  terms  of  K  and  of  yw, 
of  all  the  common  electrical  and  magnetic  quantities ;  it 
gives  also  the  ratio  of  the  dimensions  in  terms  of  K  to  the 
dimensions  in  terms  of  /*. 

In  the  table,  the  symbols  and  names  recommended  by 
the  Committee  on  Notation  of  the  Chamber  of  Delegates  of 
the  International  Electrical  Congress  of  1893  are  used,  this 
system  having  been  adopted  throughout  the  book. 

174.  Proposed  Nomenclature. — The  table  contains  the 
names  and  symbols  of  the  quantities  in  common  use; 
names  have  been  proposed  for  the  ratios  of  Flux/Force 
and  Flux-Density/Force-Intensity  for  the  dielectric  flux  to 
correspond  to  those  already  in  use  for  the  conductive  and 
magnetic  fluxes. 

For  the  conductive  flux  : 

Flux/Force  =  i/e  =  Conductance  {g)  =  I/Resistance  (r). 

Flux-Density/Force-Intensity  =  i/T  -z-  e/l  =  Conductiv- 
ity (y)  =  I/Resistivity  (p). 

For  the  magnetic  flux  : 

Flux/Force  =  $/JF  =  Permeance  =  i/Reluctance  ((R). 

Flux-Density/Force-Intensity  =  (B/OC  —  Permeability  (/*) 
=  i/Reluctivity  (v). 

For  the  dielectric  flux  two  systems  of  names  have  been 
proposed,  the  first  by  Oliver  Heaviside  and  the  second  by 
F.  E.  Nipher.  Heaviside's  *  notation  is  as  follows  : 

Flux/Force  =  N/e  —  Permittance  =  i/Elastance. 

*  Electromagnetic  7^heory,  Vol.  I. 


ELECTRICAL   AND    MAGNETIC   UNITS. 


Ratio  of 

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3! 

"i         ^          1^1"                  7"  7            7 

tential. 
^raday  Tubes, 
agnetic  Force. 

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1           1           1    1      -  1                               1 

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Equation. 

|?  II    ||     II         II   7     II     II         II     H                     II      II     "    Jl         II    II    II    II         II    II 

;  Electromotive 
y;  Magnetizing 

T  Coefficie 

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is 

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•§:ll  i  111  nil  iisg  Jin  si 

1  Electric  Field  Strength 
a  Magnetic  Field  Strengt 
8  Electric  Potential  ;  Vol 

WCL.H2      W2Q^      tLJ2o)d<      CudiCtiU     D^C^CJU     >Su 

254  UNITS  AND   DIMENSIONS. 

Flux-Density/Force-Intensity  =  D/F  =  Permittivity  = 
i/Elastivity. 

Permittance  is  capacity  ;  Permittivity  is  specific  inductive 
capacity. 

Nipher's  *  notation  is  as  follows  : 

Flux/Force  =  N/e  —  Perviance  =  i/Diviance. 

Flux-Density/Force-Intensity  =  Perviability  =  i/Divia- 
bility. 

Perviance  is  capacity ;  perviability  is  specific  inductive 
capacity. 

Neither  of  these  last  sets  of  names  is  in  use. 

175.  Dimensions  of  K  and  /*.-— The  table  shows  that  the 
dimensions  of  the  unit  of  every  quantity  in  the  AT-system 
are  related  to  its  dimensions  in  the  /^-system  by  a  simple 
function  of  LT~\K^,  in  which  Yl/does  not  enter,  namely, — 
the  quantity,  its  reciprocal,  its  square,  and  the  reciprocal  of 
its  square.  The  dimensions  thus  deduced  are  entirely  inde- 
terminate so  long  as  K  and  yw  have  no  dimensions  in  LMT. 
To  give  K  and  yw,  coefficients  depending  on  the  nature  of 
the  medium,  dimensions  in  LMT,  implies  a  mechanical 
theory  of  electricity  and  magnetism. 

If  the  phenomena  of  electricity  and  magnetism  are  differ- 
ent manifestations  of  mechanical  action  of  one  kind  or  an- 
other, then  the  quantities  AT  and  /i  must  have  dimensions  in 
LMT,  and  the  expression  of  the  dimensions  of  any  quan- 
tity, as  q,  in  these  two  systems  must  be  identical;  that  is, 
their  ratio,  a  function  of  LT~*(K)ty  must  be  zero  dimen- 
sions, or  LQM°T°\  or, 

(ATyw)"3  =  LT~l  =  v,  a.  velocity. 

Maxwell,  in  his  electromagnetic  theory,  which  does  as- 
sume all  the  phenomena  of  electricity  and  magnetism  to  be 
*  Electricity  and  Magnetism. 


ELECTRICAL   AND   MAGNETIC   UNITS.  255 

manifestations  of  ordinary  mechanical  forces,  shows  that 
the  velocity  of  propagation  of  electric  disturbances  in  a 
medium  of  specific  inductive  capacity,  K,  and  permeability, 
M,  is  (K^)-*. 

The  ratio  of  the  dimensions^being  in  all  cases  a  function 
of  F(Ajw)*,  any  values'of  K  and  /*  that  make  this  expres- 
sion unity  will  result  in  identical  dimensions  in  the  two  sys- 
tems for  every  quantity.  Any  values  of  K  and  yw  that  are 
based  on  the  assumption  of  a  mechanical  theory  of  elec- 
tricity will  fulfil  this  condition. 

Many  different  systems  of  dimensions  have  been  pro- 
posed of  late,  all  based  on  certain  assumptions  for  the 
values  of  K  and  /* ;  the  number  that  can  be  devised  is  with- 
out limit,  but  as  there  is  no  reality  behind  the  assumptions 
it  is  useless  to  record  them. 

The  two  systems  of  units  ordinarily  met  with  are  the 
Electrostatic  (E.  S.)  and  the  Electromagnetic  (E.  M.) :  the 
first  is  based  on  the  assumption  that  the  specific  inductive 
capacity  of  vacuum  is  unity,  and  is  therefore  deduced  from 
the  AT-sy stem  by  making  K  unity  in  all  the  formulae;  the 
second  is  based  similarly  on  the  assumption  that  the  per- 
meability of  vacuum  is  unity,  and  is  therefore  deduced  from 
the  yu-system  by  making  ^  unity  in  all  the  formulae. 

These  two  systems  are  the  only  ones  that  are  in  use,  and 
of  these  the  Electrostatic  system  has  no  practical  impor- 
tance ;  all  electrical  measurements  as  ordinarily  made  are  in 
terms  of  the  Electromagnetic  system,  or  in  the  "  Practical 
System,"  of  which  the  units  are  merely  multiples  of  the 
units  on  the  Electromagnetic  system. 

The  values  of  unity  for  both  K  and  /f  were  chosen 
merely  for  simplicity;  they  assume  vacuum  as  the  standard 
medium,  and  ignore  the  physical  character  of  the  two  quan- 
tities ;  that  is,  they  assume  them  to  be  merely  numerical 
coefficients.  As  simultaneous  values,  they  do  not  satisfy 


256  UNITS  AND   DIMENSIONS. 


the  condition  (K^)-*  =  velocity,  and  hence  do  not  lead  to 
identical  dimensions. 

176.  Value  of  Ratio  "z/."  —  The  ratio  of  the  dimensions 
of  any  unit  in  the  Electrostatic  system  to  its  dimensions 
in  the  Electromagnetic  system  —  that  is,  the  function  of 
V(K}*)*  —  now  reduces  to  a  function  of  "#";  this  "v"  is  a 
definite  concrete  velocity,  and  its  numerical  value  can 
readily  be  found  by  determining  the  value  of  any  electric  or 
magnetic  quantity,  first  in  one  system  and  then  in  the 
other,  the  ratio  of  the  numerical  values  representing  a  func- 
tion of  "v."  This  velocity  "v"  has  been  determined  ex- 
perimentally by  comparing  the  two  measures  of  many  of 
the  electric  units.  The  best  experiments  give  for  the  value 
of  "z/"  very  nearly  3  X  IO10  centimetres  per  second,  —  the 
same  as  the  value  of  the  velocity  of  propagation  of  light  in 
vacua.  As  "z/,"  on  Maxwell's  theory,  is  the  velocity  of 
propagation  of  an  electric  disturbance,  these  disturbances 
are  then  propagated  with  the  velocity  of  light.  But,  all 
theory  aside,  experiment  shows  this  ratio  "z;"  to  be  practi- 
cally the  velocity  of  light,  whatever  it  may  represent  physi- 
cally. 

There  are  a  number  of  ways  in  which  the  physical  signifi- 
cance of-"?;"  can  be  illustrated;  one  of  the  simplest  is 
shown  in  the  following  ideal  experiment  : 

Assume  a  conductive  sphere,  which  is  at  the  same  time 
compressible  ;  the  capacity  of  a  sphere  is  equal  in  E.  S. 
measure  to  its  radius.  That  is, 

q  =  I  X  e, 

where  /  is  the   radius  and  e  the   potential  to  which  it   is 
charged. 

Suppose  this  sphere  to  be  connected  to  the  ground 
through  a  resistance  of  which  the  value  in  E.  S.  units  is  r, 


ELECTRICAL   AND    MAGNETIC   UNITS. 

the  sphere  will  then  discharge  and  its  potential  will  fall ; 
but  imagine  the  sphere-  to  be  compressed  at  the  same  time 
at  such  a  rate  that  the  increase  of  potential  due  to  compres- 
sion equals  the  diminution  due.-to  the  discharge. 

Then  as  e  is  constant,  a  constant  current  will  flow,  rep- 
resented by 

i  =  e/r. 

But  0  =  /  x  *, 

da     *.      edl         , 

and  dt  =  '  =  Tt  =  e/r- 

That  is,  r  —  i  ^  dl/dt. 

Or,  the  velocity  dl/dt  with  which  the  radius  of  the  sphere 
must  be  shortened  is  numerically  equal  to  the  conductance 
of  the  discharge  circuit  in  E.  S.  units. 

The  chief  use  of  these  dimensions  is  to  show  the  manner 
in  which  the  fundamental  units  enter  into  the  composition 
of  the  unit  of  any  quantity,  and  thus  show  the  changes  in 
the  derived  unit  flowing  from  given  changes  in  the  fun- 
damental units.  For  instance,  if  LMT  are  changed  to 
L ' M1 '  T' ',  the  new  E.  S.  unit  of  quantity  will  be  equal  to 


times  the  old  E.  S.  unit";  similarly  the  new  E.  M.  unit  of 
quantity  will  be  equal  to 

fL'\*(M'\* 


fflSI 


times  the  old  E.  M.  unit. 

177.  Practical  System. — In  the  E.  M.  system  the  units 
of  the  quantities  in  common  use  are  either  impracticably 


258  UNITS  AND   DIMENSIONS. 

large  or  small  for  convenient  use.  This  fact  had  led  to  the 
adoption  and  use  of  a  so-called  "  Practical  "  system,  based 
on  a  unit  of  length  equal  to  io9  centimetres,  or  an  earth's 
quadrant ;  a  unit  of  mass  equal  to  io-11  grammes,  and  a 
unit  of  time  equal  to  the  second.  That  is,  in  the  Practical 
system, 

L'  =  io9  centimetres, 

M'  =  ro~n  grammes, 
T'  =  I  second. 

From  the  table  of  dimensions  of  the  various  quantities 
the  accompanying  table  is  derived,  showing  the  relation 
between  the  values  of  the  various  units  in  this  Practical  sys- 
tem and  in  the  C.  G.  S.,  Electromagnetic  system,  for  electric 
and  magnetic  quantities. 

RELATION    BETWEEN    PRACTICAL   AND    C.  G.  S.    ELECTRO- 
MAGNETIC   SYSTEMS. 

n  Name  of  Unit  Practical  Unit 

«ua  in  Practical  System.  equals 

Length Quadrant  io9  centimetres 

Mass io~n  grammes 

Time.  . Second  10°  seconds 

Area io18  cm2 

Velocity io9  cm/s 

Force ....  io~2  dyne 

Work Joule  io7  erg 

Power   Watt  io7  erg/s 

Resistance Ohm  io9  cm/s 

Current Ampere  io-1  C.  G.  S. 

Quantity Coulomb  io-1  C.  G.  S. 

Electromotive  force Volt  io8  C.  G.  S. 

Magnetic  flux ....  io8  Weber 

Magnetic  flux-density io-10  Gauss 

Reluctance io~9  Oersted 


ELECTRICAL   AND    MAGNETIC   UNITS.  2$$ 

0  Name  of  Unit  Practical  Unit 

yuantlty-  in  Practical  System.  equals 

Permeance IO9  C.  G.  S. 

Magnetic  force io~10  C.  G.  S. 

Magnetomotive  force  ....     »  A  .  io-1  Gilbert 

Capacity ... . . .     Farad  io~9  C.  G.  S. 

Inductance Henry  io9  C.  G.  S. 

Specific  inductive  capacity     ....  io~18  C.  G.  S. 

178.  Nomenclature  of  Practical  Units.  —  The  more 
common  units  in  the  Practical  system  have  long  had  the 
names  of  distinguished  men  of  science,  such  as  Ohm, 
Volta,  Ampere,  Coulomb,  Joule,  Watt,  Faraday.  These 
names  have  answered  all  ordinary  purposes  until  the  last 
few  years  ;  but  now  a  demand  has  arisen,  owing  to  the 
more  common  use  of  the  quantities,  for  names  for  the  units 
of  flux,  induction,  magnetomotive  force,  reluctance,  induc- 
tance, and  some  others.  The  matter  has  been  discussed 
widely  at  electrical  congresses  and  before  electrical  socie- 
ties. 

The  units  of  the  Practical  system  were  selected  to  be  of 
convenient  size  for  the  common  electric  quantities  ;  they 
serve  fairly  well  for  this  purpose,  with  the  exception  of  the 
Farad,  which  is  about  one  million  times  too  large,  and 
which  is  replaced  in  practice  by  the  microfarad.  But  the 
Practical  system  does  not  give  magnetic  units  of  convenient 
magnitude  ;  for  instance,  the  unit  of  flux  would  be  io8 
C.  G.  S.  units,  which  is  about  one  hundred  times  too  large ; 
the  unit  of  flux-density  or  induction  would  be  io~10  C.  G.  S. 
units,  whereas  the  C.  G.  S.  unit  itself  is  more  nearly  right ; 
the  unit  of  reluctance  is  io~9  C.  G.  S.  units,  whereas  the 
C.  G.  S.  unit  is  again  approximately  right.  The  latest  official 
body  to  pass  upon  the  question,  the  International  Electrical 
Congress  of  1893,  refused  to  adopt  new  magnitudes  or 
names  for  the  magnetic  units,  but  recommended  the  adop- 


260  UNITS  AND   DIMENSIONS. 

tion  of  the  C.  G.  S.  system.  This  same  congress,  however, 
adopted  the  Henry  as  the  unit  of  induction  in  the  Practi- 
cal system,  and  defined  it  as  follows  :  "  As  a  unit  of  induc- 
tion the  henry  is  recommended,  which  is  the  induction  in  a 
circuit  when  the  electromotive  force  induced  in  this  circuit 
is  one  international  volt,  while  the  inducing  current  varies 
at  the  rate  of  one  ampere  per  second."  It  may  be  con- 
sidered official. 

Since  this  date,  however,  the  American  Institute  of  Elec- 
trical Engineers  has  adopted,  "  provisionally,"  the  names 
Weber,  Gauss,  Gilbert,  and  Oersted  for  the  C.  G.  S.  units  of 
flux,  flux-density,  magnetomotive  force,  and  reluctance,  re- 
spectively, as  shown  by  the  table.  It  adopts  Gauss  for  the 
unit  of  magnetic  force  as  well  as  for  magnetic  flux-density; 
these  two  quantities  are  different  physical  magnitudes,  for 
which  the  same  name  should  not  be  used. 

These  names  have  not  been  generally  accepted,  and 
should  be  considered  merely  provisional.  The  need  for 
such  names  is  in  many  cases  not  clear ;  the  giving  of  names 
of  individuals  to  C.  G.  S.  units  is  not  in  conformity  with 
previous  usage  in  the  matter. 

The  British  Association  has  recently  proposed  a  different 
system  of  names  and  magnitudes.  For  unit  flux  it  recom- 
mends io8  C.  G.  S.  units,  and  calls  it  a  Weber,  thus  agreeing 
in  name  only  with  the  American  Institute  of  Electrical 
Engineers  ;  for  unit  magnetomotive  force,  to  be  called  the 
Gauss,  two  magnitudes  are  suggested,  one,  ^.nni,  and  the 
other,  ni,  or  the  ampere-turn.  There  are  other  recom- 
mendations of  minor  importance. 

179.  "Rational"  System. — Another  system  of  units  dif- 
fering fundamentally  from  these  has  been  developed  by 
Oliver  Heaviside,  and  christened  by  him  the  "  Rational  " 
System. 


ELECTRICAL   AND    MAGNETIC   UNITS.  26  1 

The  underlying  assumption  in  the  E.  S.  and  E.  M.  systems 
is  that  the  action  takes  place  directly  between  unit  quanti- 
ties of  electricity  or  unit  poles  of  magnetism  ;  they  ignore 
the  existence  of  the  medium  "entirely.  Heaviside's  system 
is  based  on  the  assumption  thaj^all  the  phenomena  are  mani- 
festations of  stresses  and  strains  in  a  medium  ;  the  medium 
is  therefore  brought  into  evidence.  This  plan  eliminates 
the  factor  471  from  the  expression  of  magnetomotive  force, 
making  it  correspond  to  ampere-turns  ;  this  is  merely  an 
incidental  advantage  of  the  "  Rational  "  system,  but  it  has 
been  seized  upon  as  the  main  reason  for  its  adoption. 

Coulomb's  law  for  the  force  existing  between  two  quanti- 
ties m  and  m'  at  distance  /, 


or,  in  the  E.  M.  system, 

f=  mm'/l\     j4=  I, 
Heaviside  puts 

OC  =  w/(4*O» 
and  /=  X,mf  =  mm'  f^itT. 

That  is,  instead  of  making  the  flux  from  a  pole  of  strength 
m  equal  to  ^itm,  he  makes  it  equal  to  m. 

The  question  is  simply,  What  is  the  rational  or  natural 
method  of  measuring  the  strength  of  a  source  at  a  given 
distance  from  its  centre  ?  Coulomb's  formula  follows  the 
ordinary  astronomical  or  gravitational  method  ;  but  this  is 
clearly  illogical  :  the  strength  of  a  source  m  at  a  distance  /  is 
the  quantity  of  m  per  unit  surface  at  that  point.  The  total 
flux  from  the  source  is  necessarily,  by  the  principle  of  con- 
tinuity, the  quantity  at  the  source.  Its  strength  at  distance 
/,  when  the  law  of  variation  is  that  of  the  inverse  square, 
is  w/47T/2,  or  the  strength  of  the  field  ;  the  force  which  the 


262  UNITS  AND   DIMENSIONS. 

field  exerts  on  another  source  m'  is  equal  to  the  product  of 
m'  by  the  strength  of  the  field  at  the  point,  —  that  is, 


mm 


This  simply  means  in  ordinary  language  that  unit  pole 
sends  out  one  line  of  force,  not  ^.n  lines. 

A  similar  irrationality  would  be  introduced  into  the  geo- 
metrical measures  if  unit  area  were  defined  to  be  the  area  of 
a  circle  whose  diameter  is  unity,  instead  of  the  area  of 
a  square  whose  side  is  unity.  If  this  were  done,  the  expres- 
sion for  areas  and  volumes  of  rectangular  surfaces  and  solids 
would  involve  ^.n  as  a  factor,  while  ^n  would  be  eliminated 
from  the  expression  for  area  of  circle  and  volume  of  sphere, 
where  they  properly  belong. 

The  changes  that  this  makes  in  the  various  units  are 
readily  obtained  by  giving  to  //  the  value  4^  in  the  table 
showing  the  dimensions  in  the  w-system. 

Denoting  units  in  the  Rational  system  by  the  subscript 
"  r,"  then  these  relations  are  found  : 


(47T)  =  rr/r  =  Lr/L  =  <R/(Rr  =  c/cr. 

Furthermore,  I  =  Wr/W=  Pr/Pt  since  the  expression  for 
energy  does  not  involve  /*. 

This  gives  the  relations  of  the  C.  G.  S.  units. 

In  passing  to  a  new  Practical  system,  either  the  same  or 
new  powers  of  ten  could  be  used.  Heaviside  seems  to  prefer 
the  use  of  a  single  multiple,  io8  or  io~8;  this,  however,  has 
the  great  disadvantage  of  changing  the  units  of  energy  and 
power  ;  it  seems  therefore  preferable  to  adhere  to  the  same 
factors  in  passing  from  the  Rational  C.  G.  S.  to  the  new 
Practical  system.  On  this  assumption, 


ELECTRICAL  AND   MAGNETIC   UNITS.  263 

"  Rational  "  Ohm  =  4^  x  Ohm 

"  Ampere  —  (4^)"*  X  Ampere 

Volt     t  c=  (47r)*  X  Volt 

"  Watt  on  Watt 

"  Weber  =  (4^)*  X  Weber 

Gilbert  =  (4*)**  X  Gilbert 

"  "  =  Rational  Ampere-turn 

There  has  been  considerable  agitation  recently  in  Eng- 
land looking  to  the  introduction  of  these  units  into  practi- 
cal use.  The  changes  suggested  are  great,  but  not  nearly 
so  radical  as  were  those  involved  in  the  introduction  of 
the  decimal  system.  The  numerical  factors  would  prove 
troublesome  in  workshop  practice  for  a  time,  but  this 
would  soon  pass  away,  and  a  simpler  and  better  system, 
and,  above  all,  a  logical  one,  would  remain. 

180.  Electrical  Standards  of  Measure. — The  various 
units  have  all  been  defined  in  terms  of  the  C.  G.  S.  system  ; 
the  physical  properties  on  which  these  definitions  rest  are 
such  as  do  not  readily  lend  themselves  to  physical  measure- 
ments ;  concrete  standards  based  directly  on  these  defini- 
tions would  not  be  capable  of  quick  and  easy  reproduction 
and  of  ready  comparison  at  all  places  and  times.  Hence 
certain  concrete  standards  that  do  meet  these  conditions 
have  been  chosen  for  some  of  the  units  ;  these  have  been 
measured  in  terms  of  C.  G.  S.  units,  by  the  most  accurate  and 
refined  methods  and  at  the  hands  of  the  most  careful  experi- 
menters. 

Resistance. — In  the  C.  G.  S.  E.  M.  system  resistance  is  de- 
fined as  the  ratio  of  electromotive  force  to  current,  both  of 
which  in  turn  are  referred  back  to  the  definition  of  unit 
pole  ;  the  concrete  standard  of  resistance  has  been  a  mer- 


264  UNITS  AND   DIMENSIONS. 

cury  column  of  certain  length  and  cross-section  ;  this  was 
first  used  in  the  form  of  the  Siemens  Unit,  as  a  more  or  less 
arbitrary  standard  ;  the  value  given  to  the  length,  100  cm., 
was  not  intended  to  reproduce  the  ohm  exactly. 

The  Electrical  Congress  of  1881  defined  the  ohm  to  be 
the  resistance  at  o°  C.  of  a  column  of  mercury  of  uniform 
cross-section  of  one  square  millimetre,  but  left  the  exact 
length  to  be  fixed  by  subsequent  experiment.  Many  ex- 
perimental researches  were  made  to  fix  this  length,  all  giv- 
ing values  approximately  106  cm.  So  the  Paris  Congress 
of  1884  adopted  this  value  under  the  name  Legal  Ohm. 
The  British  Association  in  1886  endorsed  this  value,  to  hold 
for  a  period  of  ten  years.  Later  and  more  accurate  meas- 
urements have,  however,  shown  that  this  value  is  too  small 
and  that  the  correct  value  is  106.3  cm.,  to  within  about 
Tnnnr-  This  value  was  adopted  by  the  International  Con- 
gress of  1893  at  Chicago  under  the  name  International 
Ohm. 

An  earlier  standard,  adopted  by  the  British  Association 
in  1864,  is  the  British  Association  Unit,  abbreviated  to 
B.  A.  U. 

The  values  of  these  various  units  in  terms  of  mercury  are 
as  follows : 

Siemens  Unit , 100  cm. 

B.  A.  U...    , 104.8 

Legal  Ohm 106 

International  Ohm 106. 3 

In  terms  of  the  International  Ohm  : 

Siemens  Unit 9408  Int.  Ohm. 

B.  A.  U 9863  Int.  Ohm. 

Legal  Ohm 9972  Int.  Ohm. 

Int.  Ohm i.ooo    Int.  Ohm. 


ELECTRICAL  AND   MAGNETIC   UNITS.  26$ 

Current. — -Unit  current  is  the  Ampere,  defined  to  be  one 
tenth  of  the  C.  G.  S.  unit  ;  its  concrete  representation  is 
obtained  in  the  form  of  a  certain  weight  of  silver  deposited 
by  the  current  in  a  silver  voltameter,  constructed  in  accord- 
ance with  certain  definite  specifications. 

The  most  accurate  determination  of  this  constant  gives 
the  weight  of  silver  deposited  by  one  ampere  in  one  second 
as  1.118  milligrammes.  This  definition  was  adopted  by  the 
International  Congress  of  1893,  under  the  name  Interna- 
tional Ampere. 

Electromotive  Force. — The  unit  is  the  Volt,  defined  to  be 
io8  C.  G.  S.  units  ;  it  is  represented  by  the  electromotive 
force  of  a  certain  form  of  Clark  cell  prepared  in  accordance 
with  certain  specifications.  The  yf  Jf  part  of  the  electro- 
motive force  of  this  cell,  at  a  temperature  of  15°  C.,  was 
adopted  by  the  1893  Congress  under  the  name  International 
Volt. 

181.   Recommendations  of  Congress  of  1893.  —  The 

same  Congress  adopted  these  definitions  : 

International  Coulomb,  the  quantity  of  electricity  trans- 
ferred by  an  International  Ampere  per  second. 

International  Farad,  the  capacity  of  a  condenser  charged 
to  one  International  Volt  by  one  International  Coulomb. 

Joule,  the  unit  of  work,  equal  to  io7  C.  G.  S.  units  and 
represented  by  the  energy  expended  in  one  second  by  one 
International  Ampere  on  one  International  Ohm. 

Watt,  the  unit  of  power,  equal  to  io7  C.  G.  S.  units  of 
power  and  represented  by  the  work  done  at  the  rate  of  one 
joule  per  second. 

Henry,  the  unit  of  induction,  the  induction  in  a  circuit 
when  the  electromotive  force  induced  in  the  circuit  is  one 
International  Volt,  while  the  inducing  current  varies  at  the 
rate  of  one  International  Ampere  per  second. 


ELECTROMAGNETIC   INDUCTION. 

LAWS   OF  ELECTROMAGNETIC   INDUCTION. 

182.  Induced  Currents.— Faraday  discovered,  in  1831, 
a  series  of  phenomena,  called  phenomena  of  electromagnetic 
induction,  consisting  in  the  production  of  currents  in  con- 
ductors which  are  displaced  in  a  magnetic  field  or  placed  in 
a  variable  magnetic  field. 

The  circumstances  which  give  rise  to  induction  currents 
may  be  summed  up  as  follows : 

I.  Suppose  two  neighboring  circuits  a  and  b,  such  as,  e.g., 
two  concentric  solenoids.  When  a  current  starts  in  a,  a 
current  in  the  opposite  direction  passes  through  b. 

The  first  current  is  called  the  inducing,  the  second  the 
induced,  current :  when  the  inducing  current  has  reached  a 
constant  strength,  the  induced  current  ceases.  If  the  in- 
ducing current  gradually  diminishes  to  zero,  a  new  induced 
current  is  produced  in  the  circuit  b,  this  time  in  the  same 
direction  as  the  inducing  current  and  ceasing  with  it.  These 
two  induced  currents  are  respectively  called  the  inverse- 
secondary  or  make-induced  current,  and  the  direct-second- 
ary or  break-induced  current. 

Now,  bring  near  to  a  conductor  b  a  conductor  a,  trav- 
ersed by  a  constant  current;  b  becomes  the  seat  of  an 
inverse-secondary  current  which  lasts  as  long  as  the  move- 
ment. If  we  move  a  away,  a  direct-secondary  current  arises 
in  b. 

To  sum  up,  there  is  an  induced  current  every  time  and 

266 


LAWS   OF  ELECTROMAGNETIC  INDUCTION. 


267 


as  long  as  the  strength  or  position  of  the  inducing  current 
varies. 

II.  Since  a  magnet  can  be  compared  to  a  solenoid,  it  is 
easy  to  see  that  every  variatioa  of  position  or  magnetization 
of  a   magnet  creates  induced^currents  in  a  neighboring  cir- 
cuit, the  direction   of  which  is  determined   by  substituting 
for  the   magnet  an   equivalent   solenoid  whose  position  or 
current  experiences  corresponding  variations. 

III.  Every  variation    in  the   form   of  a  circuit  or  of  its 
cu.rrent  gives  rise  to  induction  phenomena. 


FIG.  89. 

Suppose  a  circuit  containing  a  solenoid  b,  with  or  without 
an  iron  core,  Fig.  89:  a  galvanometer  inserted  in  a  shunt, g, 
shows  a  fixed  deviation,  equal  to  an  angle  <*,  when  the  cir- 
cuit is  closed  by  a  switch. 

Suppose  we  keep  the  needle  at  this  angle  by  a  stop 
which  prevents  it  from  returning  to  zero,  and  after  having 
broken  the  circuit,  close  it  again.  The  needle  is  thrown 
suddenly  beyond  the  angle  a,  which  indicates  a  current 
superadded  to  the  normal  one  in  the  galvanometer,  and 
which  may  be  considered  as  the  inverse-induced  current  of 
the  solenoid. 


268  ELECTROMAGNETIC  INDUCTION. 

Now  bring  the  needle  back  to  zero  and  keep  it  there, 
while  the  current  is  passing,  by  means  of  a  stop  on  the  side 
towards  which  it  tends  to  swing. 

At  the  moment  of  breaking  the  circuit,  we  shall  observe 
an  impulse  in  the  direction  opposite  to  a,  attributable  to  a 
direct-induced  current  developed  in  the  solenoid. 

These  induction  effects,  which  are  very  marked  when  the 
solenoid  has  a  large  number  of  turns,  and  especially  when 
furnished  with  an  iron  core,  are  called  inverse  extra-current 
or  inverse-induced  current  and  direct  extra-current  or  direct- 
induced  current.  These  extra-currents  are  clearly  shown  in 
their  physiological  effects  :  e.g.  the  difference  of  potential 
produced  by  a  single  voltaic  element  is  insensible  to  the 
touch  ;  but  if  a  circuit  be  formed  containing  a  cell  and  an 
electromagnet,  and  if  a  person  touch  the  two  terminals  of 
the  latter  with  his  fingers,  he  feels  a  slight  shock  when  the 
circuit  is  broken,  due  to  part  of  the  break-induced  extra- 
current  passing  across  the  hand. 

183.  Lenz'  Law. — Upon  analyzing  these  various  orders 
of  phenomena,  we  observe  that  every  circumstance  which 
brings  about  an  induction  current  also  brings  about  a  modi- 
fication of  the  flux  of  magnetic  force  across  the  circuit  acted 
on. 

The  case  of  induction  by  displacement  of  a  circuit  in  a 
magnetic  field  can  be  considered  as  the  counterpart  of  the 
displacement  of  a  current  under  the  action  of  a  field.  This 
analogy  has  been  investigated  by  Lenz,  who  has  given  the 
following  law  for  the  induction  produced  by  movement. 

Every  relative  displacement  of  a  circuit  and  a  field  gener- 
ates an  induced  current  which  tends  to  oppose  the  displace- 
ment by  means  of  its  electromagnetic  reaction. 

Thus,  when  a  conductor  is  displaced  in  a  field,  e.g. 
towards  the  left  of  a  figure  lying  along  the  conductor  and 


LAWS   OF  ELECTROMAGNETIC  INDUCTION.         269 

facing  in  the  direction  of  the  lines  of  force,  the  induced 
current  goes  from  the  head  of  the  figure  to  its  feet ;  the 
current  that  would  produce  the  same  displacement  would 

flow  from  its  feet  to  its  head.      .1 

f 

184.  General  Law  -of  Induction.— Lord  Kelvin  and 
Helmholtz  have  deduced  from  the  principle  of  the  conserva- 
tion of  energy  a  law  which  gives  in  every  case  the  direction 
and  strength  of  the  induced  current. 

Suppose  a  circuit,  in  a  magnetic  field,  contains  a  cell 
whose  electromotive  force  is  e ;  call  the  total  resistance  of 
the  circuit  r\  the  normal  current  is,  by  Ohm's  law, 

'  III  ii  il  i='-r-     •  :"' 

Under  these  conditions  the  energy  of  the  current  is  trans- 
formed into  heat,  so  that  in  a  time  dt 

eldt  =  /Vdt. 

But  in  consequence  of  the  reaction  of  the  field  on  the 
current,  an  electromagnetic  force  is  set  up  which  tends  to 
displace  the  current  and  make  it  do  work,  d  W,  during 
the  time  dt.  This  work  is  naturally  accomplished  at  the 
expense  of  the  cell's  energy,  being  the  only  source  of  energy 
at  hand ;  and  as  e  and  r  are  constants  by  hypothesis,  it  fol- 
lows that  the  current  has  had  to  assume  a  new  value  i  such 
that 

eidt  =  frdt  +  d  W. 

But  we  have  seen  (§  142)  that  the  electromagnetic  work 
is  equal  and  of  contrary  sign  to  the  variation  in  the  energy, 
—  t$,  of  the  current  in  the  field.  Consequently 


2/0  ELECTROMAGNETIC  INDUCTION. 

therefore 

ei&t  =  *Vd/ 

whence 

d<Z> 


We  see  that  the  term  —  -j-  plays  the  part  of  an  electro- 

motive force  which  tends  to  diminish  the  current  ;  it  is  the 
general  expression  for  the  electromotive  force  of  induction. 

If  we  discard  the  cell  and  then  displace  the  circuit  in  the 
same  way  as  before,  by  an  expenditure  of  mechanical  en- 

d<& 
ergy,  an  electromotive  force  of  induction,  —  _=  —  ,  is  set  up, 

which  produces  a  current 


z  = 


Equation  (2)  shows  that  an  immovable  circuit  traversed 
by  a  variable  magnetic  flux  is  equally  the  seat  of  an  electro- 
motive force  of  induction  which  is  in  every  case  equal  and  of 
contrary  sign  to  the  rate  of  variation  of  the  flux  with  regard 
to  the  time. 

185.  Maxwell's  Rule. — The  direction  of  the  electromo- 
tive force  is  readily  deduced  from  Maxwell's  rule  (§  135). 
If  we  suppose  a  corkscrew  to  move  forward,  rotating  in  the 
direction  of  the  field,  the  direction  of  rotation  indicates  the 
positive  direction  of  the  electromotive  force.  Now,  when 
the  flux  or  the  number  of  lines  of  force  decreases,  the  elec- 
tromotive force  —  -j-  is  positive.  If  the  flux  increases,  the 

electromotive  force   is   negative,  i.e.  it   is   in   the    opposite 
direction   to   the   movement   of   rotation  of  the  corkscrew. 


LAWS   OF  ELECTROMAGNETIC  INDUCTION.         2?  I 

To  sum  up  :  the  electromotive  force  is  in  the  direction  of  the 
rotation  of  a  corkscrew  advancing  in  the  direction  of  the  lines 
of  force,  when  the  flux  decreases ;  when  the  flux  increases, 
it  is  in  the  opposite  direction. 

Apply  this  rule  to  a  solenoid  into  which  a  magnet  enters  by 

• 

its  TV  end.  The  flux  of  force  across  the  solenoid  is  increased, 
consequently  the  electromotive  force  is  negative.  When 
the  magnet  is  drawn  out,  the  flux  decreases  and  the  electro- 
motive force  becomes  positive. 

186.  Faraday's  Rule.— In  the  case  of  induction  by  dis- 
placement, it  is  often  useful  to  determine  the  electromotive 
force  due  to  the  movement  of  the  various  parts  of  the  circuit 
acted  upon.  Now  it  will  be  remembered  (§  145)  that  the 
total  variation  of  the  flux  across  a  circuit  is  the  algebraic 
sum  of  the  elementary  fluxes  cut  by  its  various  parts. 

We  can,  therefore,  interpret  the  general  law  of  induction 
by  saying  that  the  electromotive  force  set  up  in  a  conductor 
is  measured  at  each  instant  by  the  flux  of  force  cut  in  unit 
time.  In  other  words,  the  electromotive  force  equals  the 
length  of  the  conductor  multiplied  by  the  field-intensity, 
and  by  the  projection  normal  to  the  lines  of  force  of  the 
displacement  made  in  unit  time. 

The  direction  of  the  electromotive  force  is  obtained  by 
Lenz'  law  (§  183). 

Thus  the  axle  of  a  car,  which,  in  the  northern  hemisphere, 
moves  from  east  to  west  cutting  the  terrestrial  lines  of 
force,  is  the  seat  of  an  electromotive  force  of  induction  di- 
rected from  north  to  south.  To  find  this  direction,  we  need 
only  suppose  a  figure  lying  along  the  axle  and  looking  down- 
wards (since  in  the  northern  hemisphere  the  lines  of  force 
go  downwards),  and  that  the  figure  is  displaced  to  the  left ; 
the  induced  current  then  tends  to  pass  from  the  head  to 
the  feet. 


2/2  ELECTROMAGNETIC  INDUCTION. 

Faraday's  rule  has  the  advantage  over  Maxwell's  of  apply. 
ing  even  in  the  case  of  an  open  circuit,  and  of  showing  that 
an  electromotive  force  of  induction  exists  in  conductors 
cutting  the  lines  of  force  of  a  magnetic  field. 

Fleming's  mnemonic  rule  can  also  be  used  (§  138).  In  the 
present  case  the  right  hand  should  be  used  :  The  index  finger 
and  thumb  being  respectively  in  the  direction  of  the  lines 
of  force  and  the  displacement,  the  middle  finger  will  indicate 
the  direction  of  the  induced  E.  M.  F. 

It  follows  from  the  above  rules  that  if  a  circuit  is  dis- 
placed in  a  field  in  such  a  way  that  the  flux  embraced  by  it 
remains  constant,  or  when  a  conductor  moves  parallel  to 
the  lines  of  force  in  a  field,  no  induced  currents  are  gener- 
ated. 

187.  Seat  of  the  Electromotive  Force  of  Induction.  — 

The  electromotive  force  takes  its  rise  in  all  the  parts  of  the 
circuit  which  cut  lines  of  force,  and  in  these  only.  Thus, 
in  the  above  example  of  an  axle  moving  upon  rails,  the 
E.  M.  F.  is  developed  in  the  axle. 

Suppose  that  a  magnetized  bar  be  displaced  along  the 
axis  of  a  metallic  ring.  All  the  elements  of  the  ring  cut  in 
each  instant  the  same  number  of  lines  of  force,  and  in  each 
element,  with  a  resistance  dr,  is  set  up  an  E.  M.  F.  de  and 
a  current 

.      de 


By  reason  of  symmetry  there  can  be  no  difference  of  po- 
tential in  the  ring;  but  if  the  ring  be  cut,  there  immediately 
occurs  at  the  points  of  separation  a  difference  of  potential 
representing  the  sum  of  the  electromotive  forces  of  the  dif- 
ferent elements.  This  case  shows  that  an  E.  M.  F.  can  ex- 
ist without  difference  of  potential. 

Hydrodynamics    offers   analogous    examples  :    a   circular 


LAWS   OF  ELECTROMAGNETIC  INDUCTION.         2/3 

trough  filled  with  water,  in  which  a  solid  ring  is  revolved, 
would  show  a  current  due  to  the  friction  of  the  liquid  against 
the  ring ;  the  motive  force,  being  uniformly  distributed, 
would  not  produce  any  difference  of  level  between  the  vari- 
ous parts  of  the  water  in^the  trough. 

188.  Flux  of  Force  Producing  Induction.— It  is  impor- 
tant to  determine  the  expression  for  the  flux  traversing  an 
induced  circuit. 

In  a  general  way  the  flux  can  be  separated  into  two  parts, 
the  first  due  to  the  current  itself  which  traverses  the  cir- 
cuit, the  second  to  the  external  field  produced  by  currents 
or  magnets. 

In  §  163  we  defined,  under  the  name  of  coefficient  of 
self-induction  of  a  circuit,  the  ratio  of  the  flux  traversing  it 
to  the  current.  This  coefficient  depends  on  the  form  of  the 
circuit  and  on  the  medium  in  which  it  is  placed.  In  fact, 
the  flux  of  magnetic  force  generated  is  proportional  to  the 
permeability  of  the  surrounding  medium.  If  the  circuit  is 
completely  surrounded  with  iron,  or  if  it  simply  contains  a 
core  of  iron,  the  flux  of  force  has  a  very  much  higher  value 
than  if  the  circuit  is  simply  surrounded  by  air  ;  the  perme- 
ability of  iron  for  moderate  inductions  being  very  much 
greater  than  that  of  air. 

If,  then,  we  denote  by  Lsi  the  flux  of  force  produced  by  the 
current  across  its  own  circuit,  the  coefficient  of  self-induction 
Ls  has  a  constant  value  only  on  condition  that  the  circuit  is 
invariable  in  form  and  placed  in  a  feebly  magnetic  medium. 
Where  the  circuit  is  near  very  magnetic  bodies,  its  coeffi- 
cient Ls  becomes  variable  with  the  current. 

We  cannot,  therefore,  specify  the  coefficient  of  self-induc- 
tion of  an  electromagnet  without  specifying  the  current 
which  traverses  the  field-coils,  together  with  the  former 
magnetic  condition  of  the  core. 


2/4  ELECTROMAGNETIC  INDUCTION. 

As  an  application  of  the  above,  let  us  take  an  annular 
solenoid,  in  which  the  thickness  of  the  section  is  negligible 
compared  to  the  diameter,  §  153. 

Denoting  by  n  the  number  of  turns  of  wire  per  centi- 
metre, measured  along  the  circular  axis  of  the  solenoid,  and 
by  s  its  cross-section,  the  internal  magnetic  flux,  for  a  cur- 
rent 2,  is  expressed  by 


This  flux  traverses  successively  the  n}  turns  of  the  sole- 
noid ;  consequently  the  total  flux  across  the  solenoid  is 
#  =  nX  3C.y,  and  the  coefficient  of  self-induction  has  the  value 


Such  a  solenoid  having  20  turns  per  unit  of  length  and  a 
section  of  100  cm2  would  have  a  coefficient  of  self-induction 
per  centimetre  equal  to  i6o,ooO7rC.  G.  S.  units  or  0.503  X  10  * 
quadrants. 

If  there  is  an  annular  iron  core  of  permeability  //  in  the 
solenoid,  the  coefficient  becomes 


The  permeability  sometimes  exceeds  3000,  which  explains 
why  an  electromagnet  produces  extra-currents  very  much 
greater  than  those  of  a  solenoid  without  a  core. 

The  preceding  expression  also  represents  the  coefficient 
of  self-induction  of  a  straight  electromagnet  of  great  length, 
if  we  neglect  the  influence  of  the  extremities. 

When  a  coil  is  wound  with  wire  doubled  on  itself,  a  gen- 
erator of  electric  energy  sets  up  in  it  helicoid  currents  of 
opposite  directions  whose  resultant  magnetic  effect  upon  an 
interior  core,  as  well  as  on  the  surrounding  medium,  is  zero. 
It  follows  that  such  a  solenoid  has  a  negligible  coefficient 


f.AWS   OF  ELECTROMAGNETIC  INDUCTION.         2?$ 

of  self-induction.  The  same  result  is  reached  with  a  single 
wire  if  the  successive  layers  are  wound  in  the  opposite  di- 
rections, supposing  that  there  are  an  even  number  of  layers 
having  each  the  same  number  of  turns. 

So,  too,  two  straight  conductors  near  together,  or  two 
wires  twisted  together,  do  not  give  rise  to  effects  of  lateral 
induction  when  they  are  traversed  by  currents  equal  and 
opposite  in  direction. 

When  the  current  passing  in  an  electromagnet  does  not 
exceed  the  value  which  corresponds  to  the  "  elbow  "  in  the 
core's  magnetism  curve,  §  57,  we  may  take  for  granted,  in 
approximate  calculations,  that  the  permeability  is  constant 
and  consequently  the  coefficient  of  self-induction  also. 

According  to  Lord  Rayleigh  this  hypothesis  always  holds 
good,  whatever  be  the  magnetization  of  the  core,  when  we 
are  treating  small  variations  of  the  magnetizing  force,  i.e.  of 
the  current. 

To  sum  up,  the  total  flux  across  an  isolated  circuit  trav- 
ersed by  a  current  i  is 

<Z>  =  Lj. 

If  the  current  varies,  an  E.  M.  F.  of  self-induction  is  set 
up,  equal  to 

d$  d  ,T  .. 

-37  =     -d/*^)' 

or  simply 

* 


if  Ls  is  constant. 

When  the  current  increases,  dt  is  positive  and  the  E.  M.  F. 
negative  ;  whence  an  inverse  extra-current.  When  the  cur- 
rent is  decreasing,  the  E.  M.  F.  is  positive  and  gives  rise  to 
a  direct  extra-current. 


276  ELECTROMAGNETIC  INDUCTION. 

If  the  circuit,  instead  of  being  isolated,  is  near  magnets 
or  currents,  to  the  flux  due  to  the  current  itself  is  added  a 
supplementary  flux,  comprising  lines  of  force  in  the  mag- 
netic field  produced  by  these  external  causes. 

In  the  case  of  currents,  we  have  defined  (§  142),  under  the 
name  of  coefficient  of  mutual  induction  of  two  circuits,  the 
ratio  of  the  flux  traversing  one  of  these  circuits  to  the  cur- 
rent in  the  other  circuit.  We  must  introduce  the  same 
condition  here  as  in  the  case  of  self-induction  ;  when  the 
permeability  of  the  medium  is  variable,  Lm  varies  with  the 
currents  under  consideration. 

Suppose  a  circuit  whose  own  flux  is  Lsi  and  which  is  near 
another  circuit  whose  coefficient  of  mutual  induction  and 
current  are  respectively  Lm  and  i'  .  The  total  flux  will  be 


The  E.  M.  F.  of  induction  will  in  this  case  be  expressed  by 
d$  d   , 


If  the  two  circuits  are  of  invariable  form,  and  if  we  sup- 
pose that  the  permeability  of  the  medium  surrounding  them 
is  constant,  Ls  and  Lm  are  constant  factors,  and  we  have 


d/       T    oV 


The  secondary  circuit  may,  moreover,  be  situated  in  a 
field  produced  by  the  earth  and  magnets.  If  we  denote  by 
3£s-}-2(m(*))  the  total  flux  set  up  by  these  causes  across  the 
circuit,  we  get  the  general  expression 

<2>  =  Lsi  +  LJ  +  Ws  + 
whence 


LAWS   OF  ELECTROMAGNETIC  INDUCTION.         2/7 

NOTE.  —  Let  us  take  up  again  for  a  moment  the  case  of  an 
annular  bobbin,  with  an  iron  core  closed  upon  itself  and 
traversed  by  currents  whose  direction  varies  periodically, 
passing  from  -|-  i  to  —  i  and  inversely.  At  every  change  of 
direction  the  magnetization  oLthe  core  is  reversed  and  the 
coefficient  of  self-induction  for  the  bobbin  takes  the  value 

Ls  =  ^$nn^s. 

But  if  the  currents  traversing  the  coils  vary  from  o  to  a 
value  i  without  assuming  negative  values,  i.e.  if  the  current 
is  simply  intermittent,  the  magnetic  filaments  formed  in  the 
core  remain  oriented  by  virtue  of  their  coercive  force,  and 
the  coefficient  of  self-induction  of  the  bobbin  has,  for  the 
currents  following  the  first,  sensibly  the  same  value  as  if  the 
core  did  not  exist.  Matters  go  on  as  if  the  core  were  a 
permanent  magnet  and  the  coefficient  of  self-induction  sim- 
ply equal  to 


When  the  core  is  not  continuous,  no  closed  filaments  are 
formed,  so  that  the  effect  of  the  coercive  force  is  consider- 
ably weakened. 

189.  Quantity  of  Induced  Electricity.  —  Suppose  the 
flux  across  a  circuit  varies  from  o  to  a  quantity  <Z>.  The  in- 
duced current  at  any  instant  is 


and  the  total  quantity  of  induced  electricity 

/*.        /•*    d$        $ 

q  =  /  fat  =    I =  —  -. 

3         t/o  «/o  r  r 


278  ELECTROMAGNETIC  INDUCTION. 

If  the  flux  then  returns  to  o,  the  quantity  of  electricity  is 

/-"=* 

it  is  equal  to  the  preceding,  and  is  displaced  in  the  opposite 
direction. 

APPLICATIONS   OF  THE   LAWS   OF   INDUCTION. 

190.  Movable  Conductor  in  a  Uniform  Field.— Suppose 
that  a  conductor  of  length  /,  such  as  the  axle  of  a  railway 
car,  is  displaced  horizontally  with  a  velocity  v.  It  cuts  the 
terrestrial  lines  of  force,  and  the  E.  M.  F.  of  induction  is, 
denoting  by  3C  the  vertical  component  of  the  magnetic  field, 


The  direction  of  this  E.  M.  F.  is  given  by  Faraday's  rule 

(§  186). 

Put  /=  150  cm, 

v  =  1666  cm  per  second, 

oe  =  0.44. 

We  get  e  =  150  X  1666  X  0.440.  G.  S.  units,  or  o.oon 

volt. 

191.  Faraday's  Disc. — If  Barlow's  wheel  is  caused  to 
rotate  (§  157),  there  arises  an  E.  M.  F.  directed  from  the 
periphery  to  the  centre  or  inversely,  according  to  direction 
of  rotation. 

Keeping  the  notation  used  in  the  above-mentioned  para- 
graph, and  denoting  by  n  the  number  of  rotations  per  sec- 
ond, and  by  <*>  the  angular  velocity  of  the  wheel, 

GO  =  2nn ; 


APPLICATIONS   OF   THE  LAWS   OF  INDUCTION.     2/Q 

the  E.  M.  F.  of  induction  is 


Faraday,  to  whom  this  experiment  is  due,  thus  discovered 
the  simplest  known  induction-machine.  We  also  are  in- 
debted to  him  for  the  following  arrangement,  which  is  the 
principle  of  the  so-called  unipolar  machines. 

Let  us  consider  the  apparatus  shown  in  Fig.  75,  and  sup- 
pose that  we  leave  out  the  cell,  joining  the  trough  directly 
to  the  middle  of  the  magnet.  If  the  conductor  is  now  made 
to  revolve,  it  becomes  the  seat  of  an  E.  M.  F.  of  induction, 
whose  value  is,  denoting  by  the  GJ  angular  velocity  of  the 
conductor,  n  the  number  of  turns  per  second,  and  m  the 
magnetic  mass  of  the  pole  which  serves  as  pivot, 

e  =  ^nm  X  n  =  2moo. 


Edlund  has  pointed  out  that,  by  leaving  the  conductor 
at  rest  and  rotating  the  magnet,  there  is  also  an  induced 
current  set  up.  This  can  only  be  attributed  to  the  de- 
velopment of  an  E.  M.  F.  of  induction  in  the  magnet  itself 
in  consequence  of  its  rotation  in  its  own  field. 

192.  Measurement  of  the  Intensity  of  the  Magnetic 
Field  by  the  Quantity  of  Electricity  Induced.  —  Suppose 
a  uniform  field  of  sufficient  extent  to  allow  a  flat  bobbin  to 
turn  upon  itself  in  the  field,  its  axis  of  rotation  being  along 
a  diameter  normal  to  the  direction  of  the  field.  The  ends 
of  the  wire  in  the  coil  are  connected  by  sliding  contacts  to 
a  ballistic  galvanometer  (§  150). 

Let  R  be  the  total  resistance  of  the  circuit,  3C  the  field- 
intensity.  If  the  bobbin,  containing  n  turns  of  surface  a,  is 
made  to  perform  half  a  revolution,  starting  from  a  position 
with  its  plane  normal  to  the  direction  of  the  field,  the  flux 


28O  .        ELECTROMAGNETIC  INDUCTION. 

traversing  it  passes  from  3£an  to  o,  then  from  o  to  — 
The  total  variation  is  23£an,  and  the  quantity  of  electricity 
induced  (§  171) 


Again,  denoting  by  a  the  swing  of  the  galvanometer,  by 
K  a  constant, 

whence 

_KaR 

2an  ' 

Weber  s  inclinometer,  which  serves  to  determine  the  inten- 
sity of  the  earth's  field,  is  based  upon  this  method. 

If,  by  small  equal  advances,  we  slide  a  test-coil  along  a 
magnet  which  it  exactly  encloses,  we  will  get  in  a  ballistic 
galvanometer  connected  with  the  test-coil,  swings  propor- 
tional to  the  field-intensity  in  the  various  regions  of  the 
magnet.  This  is  a  convenient  way  to  obtain  the  distribu- 
tion-curve of  magnetism  in  a  magnet  (§  47). 

193.  Expression  for  the  Work  Absorbed  in  Magneti- 
zation. Loss  due  to  Hysteresis. — Take  the  case  of  an 
annular  electromagnet  (§  142)  of  sufficient  diameter  to  allow 
the  supposition  that  the  internal  field  is  uniform  and  ex- 
pressed by 

3C  == 


n1  being  the  number  of  turns  per  unit  length. 

The  work  done  to  magnetize  the  electromagnet  is  neces- 
sarily equal  to  the  energy  given  back  by  it  under  the  form 
of  the  extra-current.  Now  the  energy  of  this  latter  is  given 

by 

W=ft  eiAt  =/"-        id/  -  /- 


APPLICATIONS   OF  THE  LAWS   OF  INDUCTION.     28 1 

But,  denoting  by  s  the  cross-section  of  the  magnet,  /  its 
mean  length,  and  (B  the  induction  traversing  it,  we  have 


1  3C 

Substituting  for  £>  this  valu^e,  and  for  i,  -  ,  we  get 

• 


The  energy  expended  per  unit  of  volume  will  be 

—  i  f  JCd(B. 

V 


Now  (B  =  3C 

whence 

—  —  /   3Cd&  =  —  — 
t/ 


The  integral 


represents  the  energy  restored  by  the  solenoid,  that  is  to 
say  its  intrinsic  energy. 
The  integral 


is  therefore  the  energy  restored  by  demagnetizing  the  core. 
If  the  magnetizing  force  varies  between  values  3C,lt  5C2, 
this  last  expression  of  energy  will  be 


It  would  be  represented  by  the  area  comprised  between 
the  curve  3  =/(3e),  the  axis  of  0  and  two  parallels  to  the 


282 


ELECTROMAGNETIC  INDUCTION. 


axis  of  abscissae  drawn  through  the  points  of  the  curve  cor- 
responding to  5Cj  and  3C2. 

Where  the  current  successively  assumes  values  -f-  z,  —  i, 
the  energy  restored  by  the  solenoid  equals  the  energy  ab- 
sorbed, for  we  have 


But  the  integral 


is  not  zero,  since  it  represents  the  area  comprised  between 
the  curves  AC  A'  and  AC' A  of  Fig.  90. 


FIG.  90. 

This  integral  expresses  the  loss  per  cycle  and  per  cm3  of 
the  core. 


APPLICATIONS   OF   THE  LAWS   OF  INDUCTION.     283 

The  work  expended,  equal  and   of  contrary  sign   to  the 
work  restored  by  the  core,  is  represented  by 


194.  Self-induction  in  a  Circuit  composed  of  Linear 
Conductors.  Case  of  a  Constant  Electromotive  Force. 
Time-constant. — When  a  circuit  containing  an  element 
having  constant  E.  M.  F.  is  closed,  the  current  does  not 
instantaneously  attain  its  normal  value,  especially  if  the  cir- 
cuit contains  an  electromagnet. 

In  the  same  way,  when  the  circuit  is  broken  the  current 
does  not  stop  suddenly,  but  is  prolonged  by  the  extra-current 
which  appears  in  the  spark  at  the  point  of  interruption. 
Faraday  showed  the  analogy  existing  between  these  phe- 
nomena and  those  caused  by  the  inertia  of  fluids.  A  liquid 
current  cannot  be  set  up  or  cease  suddenly  in  a  pipe,  and 
at  the  moment  of  stoppage  we  observe  a  sudden  blow  due 
to  the  momentum  of  the  moving  fluid  ;  but  there  are  pro- 
found divergences  between  these  two  cases. 

While  the  blow  given  by  a  liquid  is  diminished  by  bends 
in  the  pipe,  the  extra-current  is  much  more  marked  in  a  coil 
than  in  a  straight  wire  of  equal  length.  It  will  be  seen  fur- 
ther on  that  this  difference  is  explained  by  placing  the  seat 
of  the  current's  energy  in  the  medium  surrounding  the  con- 
ductors. 

There  exist,  however,  even  in  the  expressions  for  the  phe- 
nomena, analogies  which  it  is  useful  to  bring  out. 

Thus,  when  a  fluid  is  urged  to  move  in  a  pipe  the  force 
expended  is  utilized  on  the  one  hand  to  overcome  the  fric- 
tion against  the  walls,  on  the  other  to  increase  the  momen- 
tum of  the  movable  mass.  If  the  velocity  v  is  small,  the 
friction  can  be  expressed  by  Av,  when  A  is  a  constant. 


284  ELECTROMAGNETIC  INDUCTION. 

The  increase  of  momentum  of  the  mass  m  is  in  unit  time 


l 

m-r~.     The  total  force  is  therefore 
d/ 


(i) 


Now  take  the  case  of  a  circuit  with  a  resistance  r,  whose 
coefficient  of  self-induction  Ls  is  constant  (§  198),  and  which 
includes  a  cell  with  an  E.  M.  F.  denoted  by  E. 

At  the  moment  of  closing  the  circuit  there  is  set  up  an 

E.  M.  F.  of  induction  —  A~r»  so  that  the  current  is  given  by 
the  equation 


I  = -. 

From  this  we  deduce 

R—  ri-\-  T~  (?\ 

^a? 

an  expression  analogous  to  (i).     ri  represents,  likewise,  the 
portion  of  E.  M.  F.  used  to  overcome  the  friction  of  the 

conductor  and  Ls-r~  =  e  the  portion  used  to  increase  the  in- 
trinsic energy  of  the  circuit,  for,  multiplying  e  by  t,  we  have 

r  di  .        d 


It  is  taken  for  granted  in  equation  (2)  that  the  distribu- 
tion of  the  current  is  uniform  over  the  section  of  the  con- 
ductor, and  that  the  resistance  of  the  latter  to  a  variable 
current  is  the  same  as  to  a  steady  current.  Further  on  we 
shall  see  that  this  is  not  the  case  for  linear  conductors,  i.e. 
those  of  very  small  cross-section. 

At  the  end  of  a  time  t  the  current  will  attain  a  value  *» 


APPLICATIONS   OF   7 'HE  LAWS   OF  INDUCTION.     285 

given  by  the  integration   of   the  differential  equation  (2). 
We  have 


whence 


/*      d*  r'dt 

«/•  E-fa-tJ.  L; 


and  lastly 

£/  -1\ 

I  =  —  I  I  =  e     La], 

r  \  I 

in  which.*  represents  the  base  of  the  Naperian  logarithms. 

•p 
Theoretically,  the  current  does  not  reach  its  full  value  — 

_  rt 
until  an  infinite  time  has  elapsed,  but  as  the  value  of  e    LS 

decreases  rapidly,  this  term  becomes  negligible,  compared 
with  unity,  in  a  short  time. 

The  ratio  — ,  which  is  homogeneous  with  a  time,  is  called 

the  time-constant  of  the  circuit ;  we  will  denote  it  by  r. 

Suppose,  for  example,  an  annular  solenoid  of  100  cm2  sec- 
tion, having  20  turns  per  cm,  measured  on  the  axis,  and 
whose  length  when  developed  is  100  cm.  If  the  resistance 
be  i  ohm  or  10°  C.  G.  S.  units,  the  exponential  factor  as- 
sumes the  value 


£         47T  X  20X2000  X  100  £  l67T 

It  is  seen  that  for  a  comparatively  small  value  of  /  this 
term  may  be  neglected. 

The  variation  of  the  current  in  terms  of  the  time  is  rep- 
resented by  a  curve  which  rises  rapidly  from  the  origin, 
then  tends  towards  an  asymptote  parallel  to  the  axis  of  the 


286  ELECTROMAGNETIC  INDUCTION. 

times.     The  first  part  of  the  curve,  corresponding  to  small 
values  of  the  time,  can  be  replaced  by  a  right  line, 


Et 


This  equation  shows  that  during  the  first  instants  the 
current  depends,  not  on  the  resistance  of  the  circuit,  but  on 
its  self-induction. 

The  quantity  of  electricity  which  passes  in  the  circuit 
during  the  variable  period  is 

C* .  ,          r*Et  _*\  ,        E        E  r*     *  * 

q  =    I  idf=  /  ~~  I  -  e    7V  d*  55  —I /  e'^dt 

i/o  e/o   r  \  j  r         TVO 

E,.  E    -*- 


Beyond  a  certain  value  of  /  the  second   term  becomes 
negligible,  and  we  have  simply 


—  /  is  the  quantity  of  electricity  which  would  have  passed 
during  the  time  t  if  the  current  had  instantaneously  assumed 
its  permanent  value.  The  term  —  r  represents  the  quantity 

due  to  the  inverse  extra-current  or  make-induced  current. 

We  come  now  to  the  break-induced  current,  and,  to  sim- 
plify the  calculation,  suppose  that  the  resistance  of  the  cir- 
cuit be  maintained  constant  by  substituting  for  the  cell  a 
wire  having  the  same  resistance. 

The  break  extra-current  is  given  by  the  equation 

dt 


APPLICATIONS   OF   THE   LAWS   OF  INDUCTION.     28/ 
£ 

Integrating  between  --  and  t,  we  find 

E  _£ 

i  =  —  e  T. 
r    , 

It  is  easy  to  see  that  this  current  decreases  rapidly. 

The  quantity  of  electricity  due  to  the  extra-current  on 
breaking  is  the  same  as  that  of  the  extra-current  on  making 
the  circuit. 

The  total  quantity  of  electricity  produced  by  the  cell  is, 
moreover,  the  same  as  if  there  were  no  induction-effects, 
for 


We  can  find  directly  the  expression  for  the  quantity  of 
induced  electricity  by  applying  the  equation 

q  =  T  *          (8  189) 

and  observing  that  $  is  the  product  of  the  current,  when 
fully  established,  into  the  coefficient  of  self-induction. 

We  see  that  the  induction  phenomena  which  occur  in  a 
circuit  during  the  variable  period  of  the  current  have  the 
effect  of  causing  an  apparent  increase  in  the  resistance  of 
the  conductors.  The  total  quantity  of  electricity  put  in 
motion  is,  however,  the  same  whatever  be  the  coefficient 
of  self-induction,  for  the  direct  extra-current  restores  the 
quantity  of  electricity  abstracted  at  the  beginning  of  the 
current. 

195.  Work  accomplished  during  the  Variable  Period. 

—By  Joule's  law,  the  work  accomplished  during  the  vari- 
able period  on  closing  the  circuit  is  expressed  by 

w,= 


2  r 


288  ELECTROMAGNETIC  INDUCTION. 

If  t  is  large  enough,  we  have  simply 

E" 


r 

Now  the  cell  produces  during  the  variable  period  a  total 
quantity  of  electricity 

_  & 

This  quantity  exceeds  the  Joule  effect  by 

LJ' 


denoting  by  /the  current  when  fully  established.  This  dif- 
ference represents  the  intrinsic  energy  of  the  circuit  trav- 
ersed by  the  electric  flux  (§  144). 

We  see,  then,  that  when  a  current  is  set  up  in  a  circuit, 
the  energy  furnished  by  the  source  of  electricity  is  composed 
of  two  parts,  the  one  transformed  directly  into  heat  by  the 
Joule  effect,  the  other  stored  up  in  the  potential  state. 

This  intrinsic  energy  is  in  its  turn  transformed  into  heat 
during  the  extra-current  on  breaking  the  circuit,  for  there  is 
then  a  Joule  effect  represented  by 


/oo  772     /»oo         2t 

?rdt  =  ~        ,-7d/  = 
r  <A 


W. 


The  potential  energy  of  the  current,  according  to  modern 
ideas,  resides  in  a  special  state  of  tension  or  movement  of 
the  magnetic  field. 

It  corresponds,  as  we  have  seen,  §  194,  to  the  kinetic 
energy  of  a  mass  in  motion. 

Second  Demonstration. — The  expression  for  the  intrinsic 
energy  of  a  circuit,  measured  by  the  difference  between  the 
energy  furnished  by  the  source  and  that  absorbed  by  the 


APPLICATIONS   OF   THE  LAWS   OF  INDUCTION.     289 

Joule  effect  during  the  variable  period,  can  be  deduced  di- 
rectly from  the  general  law  of  induction 


for  we  can  write  this  in  the  form 

/t  /•*  />/  T  n 

Eidt-  /  ?rdt  =  I  Ljdi  =  —. 
«A  t/o  2 

196.  Application  to  the  Case  of  Derived  Currents.  — 

Suppose  two  conductors  joined  in  parallel,  the  self-induction 
of  the  first  being  Ls,  that  of  the  second  having  a  negligible 
value  ;  suppose  further  that  the  two  branches  have  no  mu- 
tual inductive  influence  on  each  other.  When  we  send  a 
current  into  the  two  conductors,  whose  resistances  are  rl 
and  ra,  the  division  of  electricity  between  them  is  influenced 
by  the  reaction  of  the  self-induction  in  one  of  them.  Let  il 
be  the  current  in  this  one  at  any  instant  of  the  variable 
period,  and  z'2  the  current  in  the  other  at  the  same  instant. 

The  first  one  is  the  seat  of  an  E.  M.  F.  equal  to  —  ATT, 

so  that  on  applying  Kirchhoff's  second  law  to  the  closed  cir- 
cuit including  rl  and  r2,  we  get 


whence 

&  -  r*fi&  =  -  L,fdtlt 


the  integration  being  comprised  between  corresponding  lim- 
its of  time  and  current. 

If  the  circuit  of  the  cell  is  closed  and  then  opened,  the 


2QO  ELECTROMAGNETIC  INDUCTION. 

initial  and  final  currents  are  zero  in  the  branch  r^\  conse- 
quently the  second  member 


and 


q^  and  qt  denoting  the  quantities  of  electricity  that  have 
traversed  the  two  branches. 

These  quantities  are  exactly  the  same  as  if  there  had 
been  no  induction  effect  in  the  two  branches. 

The  same  effect  would  occur  if  a  condenser  were  dis- 
charged through  the  branches.  The  division  would  occur 
as  if  there  were  no  self-induction  currents  in  one  of  them. 

It  is  taken  for  granted,  in  these  deductions,  that  the  con. 
ductors  have  a  sufficiently  small  cross-section  for  the  cur- 
rent to  be  uniformly  spread  over  every  part  of  it  during  the 
variable  period. 

197.  Discharge  of  Condenser  into  a  Galvanometer 
with  Shunt.  —  The  above  calculations  cannot  be  applied 
without  reserve  in  the  case  of  discharging  a  condenser  into 
a  galvanometer  furnished  with  a  shunt.  If  the  galvanometer 
remains  motionless  during  the  whole  period  of  discharge, 
the  subdivision  of  the  quantities  of  electricity  takes  place 
according  to  the  above  simple  law.  But  it  frequently  hap- 
pens that  the  needle  commences  to  move  before  the  end  of 
the  discharge  ;  in  which  case  it  produces  in  the  galvan- 
ometer-coil an  induced  current  opposite  in  direction,  by 
Lenz'  law,  to  that  which  would  itself  have  produced  the 
movement.  It  follows  that  the  total  quantity  of  electricity 
which  traverses  the  coils  is  diminished. 

To  calculate  this  diminution  we  may  suppose,  with  M.  L. 
Clark,  that  the  flux  of  magnetic  force  produced  by  the 


APPLICATIONS   OF  THE  LAWS  OF  INDUCTION.     291 

needle  across  the  coils  is  proportional  to  the  sine  of  the 
angle  described. 

Denote  by  i^,  g,  Ls  the  current  in  the  galvanometer,  its 
resistance  and  coefficient  of  sejf-induction  ;  by  za,  s,  the  cur- 
rent in  the  shunt  and  its  resistance,  and  suppose  it  to  be 
formed  of  a  straight  wire  or  bobbin  with  double  winding. 
KirchhofFs  second  law  shows  that 

T  dil   .    ^d  sin  a 
v-t£=LsW+K-^r, 

whence 

sJ9  'i*dt  -  g£i&  =  Lsfo  di,  +  K^"d  sin  a. 

The  current  in  the  galvanometer  is  zero  at  the  beginning 
and  end  of  the  discharge,  whose  duration  is  /,  and  the  supe- 
rior limit  of  OL  is  the  deviation  tf  of  the  needle. 

Let 


we  will  have 


But  by  the  theory  of  the  ballistic  galvanometer  (§  150),  if 
the  arc  of  swing  is  small  enough 


PL 

sin  d  =  2  sin  —  = 


a  being  a  constant. 
Consequently 


Denoting  by  Q  the  total  discharge  equal  to  ql  -\- 

S(Q  —  <?i)  —  gq±  —  aKq^  —  o, 
whence 


2Q2  ELECTROMAGNETIC  INDUCTION. 

198.  Self-induction  in  a  Circuit  of  Linear  Conductors 
where  there  is  a  Periodic  or  Undulatory  Electromotive 
Force.  —  From  a  practical  point  of  view  a  very  important 
case  of  induction  occurs  when  one  or  more  spirals  of  wire  are 
rotated  in  a  magnetic  field. 

For  simplicity's  sake,  suppose  a  coil  abgd  turns  about  the 
axis  bd  (Fig.  85)  in  a  direction  opposite  to  the  movement  of 
the  hands  of  a  watch,  in  a  uniform  field  whose  direction  is 
normal  to  the  plane  of  the  coil  in  its  present  position.  Dur- 
ing a  semi-revolution  the  part  bgd  will  cut  the  lines  of  force 
in  one  direction,  the  part  dab  in  the  opposite.  The  E.  M.  Fs. 
produced  will  be  added  together  and  produce  an  induction 
current  in  the  coil.  During  the  next  half-revolution  the 
direction  of  displacement  of  the  two  halves  of  the  coil  will  be 
reversed  and  consequently  the  induced  current  will  change 
its  direction  in  the  coil. 

In  one  complete  revolution,  the  current  is  therefore  re- 
versed twice  in  the  coil.  It  is  easy  to  see  that  the  E.  M.  F. 
is  maximum  at  the  moment  when  the  lines  of  force  are  cut 
normally,  i.e.,  when  the  plane  of  the  coil  is  parallel  to  the 
direction  of  the  field. 

Let  us  apply  the  general  law  of  induction  to  this  case. 

Let  3C  be  the  field-intensity,  S  the  surface  of  the  coil, 

a  =  -^  its  angular  velocity,  supposed  to  be  constant.     The 

angle  described  in  a  time  t  will  be  a  =  at. 

When  the  plane  of  the  coil  makes  an  angle  a  with  the 
direction  normal  to  the  field,  the  flux  traversing  the  coil  is 


cos  a  = 
The  E.  M.  F.  due  to  the  field  is 


E  =  -  -        =  S3C  sin  c-  =  S&a  sin  a  =  S3C  sin  at. 
dz  cu 


APPLICA  T1ONS    OF   THE   LA  WS   OF  IN D  UCTION.     293 


Let  T  be  the  period,  that  is,  the  duration  of  one  complete 
revolution,  corresponding  to  an  angle  2?r,  and  n  the  fre- 
quency or  number  of  periods  per  second.  The  number  of 


FIG.  91. 

alternations  is  double  the  number  of  periods.     We  can  write 
indifferently 

E  —  S3£a  sin  at  =  S3C#  sin—/  =  SW,a  sin  2nnt. 


For  the  values 


we  have 


T     3T     $T 

~4*      4'      4' 


sin—/  =  +  i,  —  I,  +  I. 

The  E.  M.  F.  passes  successively  through  maxima 

EQ  =  S3£a 
and  through  minima 

—  E0  =  —  SOCtf . 


294  ELECTROMAGNETIC  INDUCTION. 

The  expression  representing  the  variation  of  the  E.  M.  F. 
as  a  function  of  the  time  can  therefore  be  put  in  the  form 

E  =  E0  sin  at  .......     (i) 

The  current  in  the  coil  results  from  the  combination  of  the 
E.  M.  F.  due  to  the  field,  and  the  E.  M.  F.  produced  by  the 
self-induction  Ls  of  the  coil,  whose  resistance  is  r  ;  conse- 
quently, if  Ls  is  constant,  we  have 

r.         r  d*         „      .  r  dt 

E-L,-      E^at-L,-t 


whence 


di  +  ~idt  =  y  sin  atdt,       .     .     .     .     (3) 


If  we  put  t  =  uv,  u  and  v  being  arbitrary  variables,  we 
get 

K(dz>  +  ^d/)  +  v^u  —  j^  smatdt.       .    .    (4) 
\  L,s       I  L>s 

Now  make 

dv  -\--j-vdt  =  o, 
Ls 

and  for  simplicity  put  -j-  =b,  when  we  get 


logeK  being  a  constant  of  integration  ;  whence 

v  =  Ke~ftdt  .......     (5) 

Now  equation  (4)  reduced  to 

E 

vdu  =     ^  s 


APPLICATIONS  OF  THE  LAWS   OF  INDUCTION.     2$$ 
gives 


d/  .        ... 

du  =  -^ej      -=£  sin  atdt  ; 
A  L,s 


whence 


u  =  sin  atdt; 

•*•    A 


and  putting 

KK'  =  D, 


f*-  si"  atdt  +  D  \ 


=  /-  sn 


Integrating  by  parts,  we  get 

/ 

whence 


sn  ^//  =    a   ,    t,     sn  ^/  —  «  cos 


,'  =         f  "       (^  sin  tf/  -  «  cos  a*)  +  ^-«.    .    (6) 

•^sV"1     "I      ^  / 

We  can  substitute  for  the  difference 

a 
sin  at  --  cos  ^f 


by  sin  (^/  —  0),  the  value  of  0  being  determined  by  the 
condition  that  the  equality 

sin  at  --  .  cos  at  =  sin  (at  —  0) 

* 


be  verified  for  all  the  values  of  /. 
Now  for 


t=o  we  get     sin  <p  = 


296  ELECTROMAGNETIC  INDUCTION. 

for 


7T  , 

=  —  ,  0050— 

2*' 


therefore 


The  term  De~bt  of  equation  (6)  expresses  the  increase  of 
the  current  during  the  first  moments  the  E.  M.  F.  is  act- 
ing. After  a  few  instants  this  term  becomes  negligible, 
and  the  current  is  given  by 


Substituting  for  a  and  b  their  values,  we  get  finally 

E. 


(8 


on  the  condition  that 

0  =  tan-^  .......    (9) 

We  could  also  give  equation  (8)  the  form 

E,  (27tt  \ 

l  —        _          _  sm        --  0 
rVi  +  tan'  <f>        ^   T  ' 

E.  COS   0     .       /27T*  A 

-    --  sm  -  <>}  (10) 


Equation  (10)  shows  that  the  maximum  current  is 
E0  cos  0 


_ 


APPLICATIONS   OF   THE  LAWS   OF  INDUCTION. 

It  is  observed  that  the  E.  M.  F.  of  self-induction  reduces 
the  maximum  current,  which  would  be  — °  if  Ls  were  zero. 

E0  cos  0  represents  the  maximum  effective  E.  M.  F.  re- 
sulting from  the  combination  Q£  the  electromotive  force  E0 
and  the  reaction  of  the  self-induction.  The  subtractive  term 
0,  which  is  not  found  in  equation  (i)  of  E.  M.  F.,  shows 
that  there  is  a  retardation  of  phase  between  the  maximum 
values  of  the  current  and  the  E.  M.  F.  due  to  the  field. 

T 

This  retardation  is  — 0  in  duration. 

Remarks. — I.  These  results  are  applicable  to  a  coil  com- 
posed of  a  number  of  turns,  and,  for  approximate  calcula- 
tions, to  a  coil  with  an  iron  core,  provided  the  magnetiza- 
tion be  sufficiently  feeble  to  allow  us  to  take  the  permeabil- 
ity as  constant,  without  much  error. 

II.  We  denote  by  the  name  of  apparent  resistance  of  a 
circuit  the  radical 


by  which  the  E.  M.  F.  must  be  divided  in  order  to  find  the 
current  :  Heaviside  has  called  this  the  impedance  of  the  cir- 
cuit. It  will  be  noticed  that  the  radical  is  homogeneous  with 

a  resistance  and  can  be  expressed  in  ohms.     The  term  — =-* 

is  called  the  reactance. 

III.  Denoting  by  t  the  time-constant  of  the  circuit  equal 

to  — ,  the  result  can  also  be  put  in  the  form 


298 


ELECTROMA  GNE  TIC  IND  UCTION. 


These  formulae  show  that  the  apparent  resistance  (impe- 
dance) and  the  retardation  of  phase  depend  essentially 
on  the  time-constant.  A  large  self-inductance  may  only 
produce  a  minimum  apparent  increase  of  resistance,  if  the 
resistance  is  already  considerable  in  itself. 

199.  Graphic  Representations. — The  axis  Oy  represent- 
ing the  direction  of  a  uniform  field  which  develops  induc- 
tion-currents in  a  coil  turning  in  the  direction  of  the  arrow, 
let  us  represent  by  the  right  line  OM,  which  makes  an  angle 

a  =  -=-  with  Ox,  the  maximum  electromotive  force  E0.  The 

E.  M.  F.  due  to  the  field  is  represented  at  the  instant  /  by 
the  projection  oa  =  EQ  sin  a. 

The  maximum  effective  electromotive  force,  E0  cos  0,  is 
represented  by  ON,  the  projection  of  OM  on  a  straight  line 
making  an  angle  0  with  the  latter. 

Now  as  the  effective  E.  M.  F.  is  the  resultant  of  En  and 


cl--vP 


FIG.  92. 

the  E.M.  F.  of  self-induction,  the  maximum  of  this  last  will 
be  shown  by  a  right  line  OP  =  E0  sin  0,  which  completes 
the  parallelogram  OM.  NP.  The  projections  of  OM,  ON, 


APPLICATIONS   OF   THE  LAWS   OF  INDUCTION.     299 

and  OP  on  Oy  represent  the  values  of  the  different  E.  M.  Fs. 
at  the  moment  of  rotation  when  the  coil  makes  an  angle  a 
with  Ox. 

The  effective  E.  M.  F.  at  tfyis*  moment  is 

Ob  =  ON  sin  (a^—  (p)  =  7s0  cos  0  sin  (a  —  0); 

it  is  less  than  the  E.  M.  F.,  Oa,  due  to  the  field,  by  a  quan- 
tity ab  =  Oc  which  measures  the  reaction  of  self-induct- 
ance. 

But  when  the  coil  OM  has  passed  the  axis  Oy  by  an 
angle  0,  OP  comes  above  the  axis  of  x  and  the  projection 
of  the  resultant,  ON,  is  greater  than  the  projection  of  OM, 
for  the  action  of  the  self-induction  is  now  added  to  the 
E.  M.  F.  due  to  the  field. 

On  rotating  the  parallelogram  OMNP  about  the  point  O, 
the  projections  of  OM,  ON,  and  OP  on  Oy  show,  at  each 
instant,  the  relative  values  of  the  various  E.M.Fs.  in  action. 

The  current  in  the  coil  is  given  at  any  given  instant  by 
the  ratio  of  the  length  of  Ob  to  the  resistance  r  of  the  coil. 

We  can  also  show  the  variations  of  the  E.  M.  Fs.  in  terms 
of  the  time,  by  drawing  the  curves  represented  by  the 
equations 

271 't 

E  =  EQ  sin  -=-  =  E0  sin  at, 

d* 

e=  —  L,-^  =  —  E0  sin  0  cos  (at  —  0), 

E'  =  EQ  cos  0  sin  (at  —  0). 

We  will  then  get  three  sinusoidal  curves  like  those  in 
Fig.  93;  the  sine-curve  E  represents  the  E.  M.  F.  due  to  the 
field;  e  is  the  E.M.F.  of  telf-induction  and  E'  the  effective 
or  resultant  E.  M.  F.  The  ordinates  of  E'  are  the  differences 
between  those  of  E  and  e. 


300 


ELECTROMAGNETIC  INDUCTION. 


The  current  at  any  moment  is  found  by  dividing  the  or- 
dinates  of  the  curve  E'  by  the  resistance  of  the  circuit ;  the 
phase  of  the  current,  moreover,  coincides  with  that  of  E'. 


FIG.  93. 

The  retardation  of  the  current-phase  behind   that  of  the 
E.  M.  F.  due  to  the  field  is 


where 


The  maximum  possible  retardation  corresponds  to 
a.Ls 


=  oo  . 


We  have  then 


0  =  —    and     0=  — 
2  4' 


It  will  be  observed  that  the  curve  e  is  a  quarter  phase,  or 

T 

— ,  behind  the  curve  E ' .     Consequently  if  the  latter  is  itself 

4 

T 

—  behind  E,  the  positive  waves  in  E  will  be  exactly  over 

4 

the  negative  waves  in  e.     The  effective  E.  M.  F.  ought  then 

to  be  zero  (and  also  the  current)  for 
E0  cos  0=o. 


APPLICATIONS   OF   THE   LAWS   OF  INDUCTION.     3OI 

200.  Mean  Current  and  Effective  Current.  Measure- 
ment by  Dynamometer.  —  The  quantity  of  electricity  which 
passes  through  the  circuit  in  a  half-period  is  independent  of 
the  lag  ;  it  is  expressed  by 


=.    f'idt  =  "  P  si 

J^  yr'  +  a'L,'Jt 

The  mean  current,  during  a  half-period,  is 
-  9  -  2  E>         2 


/  being  the  maximum  current. 

The  mean  current-strength,  which  may  be  represented  by 

i    rkr 
T^F     MI, 


is  therefore  equal,  in  the  case  under  consideration,  to  the 
product  of  the  maximum  current  by  the  factor  — . 

It  is  to  be  observed  that  the  needle  of  a  galvanometer 
gives  no  deviation  when  the  coil  of  the  instrument  is  trav- 
ersed by  an  alternating  current  of  short  period,  for  it  then 
receives  equal  and  contrary  impulses  ;  but  it  is  possible  to 
use  the  electrodynamometer,  §  152,  whose  readings  are  pro- 
portional to  the  square  of  the  current. 

With  this  instrument  we  get  a  deviation  proportional  to 
the  mean  of  the  squares  of  the  current. 

This  mean  is  expressed  by 


302  ELECTROMAGNETIC  INDUCTION. 

and  its  square  root  is  called  the  effective  current,  the  lumin- 
ous effects  of  the  current  in  electric  lamps,  for  example, 
depending  upon  it. 

The  effective  £.  M.  F.  is  likewise  defined  by  the  square 
root  of  the  mean  square  of  the  E.  M.  F. 

The  impedance  of  the  circuit  is  the  factor  by  which  the 
effective  current  must  be  multiplied  to  obtain  the  effective 
E.  M.  F.  Sometimes  the  name  of  inductive  r:sistance  is 
given  to  a  resistance  having  a  coefficient  of  self-induction. 

In  the  present  case,  the  mean  square  of  the  current  is 


It  will  be  observed  that  the  square   root  of   the   mean 
square  is  different  from  the  mean  current, 


=  0.9; 


n*% 

consequently 

im  =  0.9  ^(TJT; 

that  is  to  say,  the  readings  by  the  electrodynamometer  must 
be  reduced  by  a  tenth  to  get  the  mean  current. 

If  we  denote  by  EQ  and  /the  E.  M.  F.  and  the  maximum 
current,  it  is  easy  to  express  in  terms  of  these  quantities 
both  the  mean  E.  M.  F.  and  current  and  the  effective 
E.  M.  F.  and  current. 


APPLICATIONS  OF   THE  LAWS   OF  INDUCTION.     303 
We  have  then  : 
Mean  E.  M.  F.  e    =     £. 


Mean  current  im  =  —  /. 


Effective  E.  M.  F.  V(?)m  =  Eeff  =  ^±. 

4/2 


Effective  current        S=  I=  -      =         ., 


Impedance  V  r*  -\ ^j^- 


It  must  not  be  forgotten  that  these  various  relations  do 
not  hold  except  in  the  case  where  the  periodic  E.  M.  F.  is 
a  simple  sine-function  of  the  time  and  where  the  self-in- 
ductance Ls  is  constant ;  which  makes  it  necessary  that  the 
permeability  of  the  surrounding  medium  be  invariable. 

If  the  periodic  function  were  more  complex,  it  might,  by 
Fourier's  theorem,  be  represented  by  a  sum  of  sinusoids, 
but  the  above  coefficients  of  reduction  would  be  changed. 

The  mean  heat  developed  by  the  current  in  one  second  is 

i  r  i  r 

Y         frdf  =»-r  X  •*'' I     «/; 

*J  0  I/  i 

that  is,  the  product  of  the  real  (or  "  ohmic  ")  resistance  of 
the  circuit  by  the  square  of  the  effective  current.     We  have 


.  =  -Ej  cos  0, 


304  ELECTROMAGNETIC  INDUCTION. 

whence 


Pm  =  \/(e*)m  V(i\  cos  0  =  Eeffleff  cos  0. 

We  see  that  the  heat  developed  in  a  circuit  where  a  peri- 
odic E.  M.  F.  is  acting  varies  with  the  lag  of  the  current- 
phase  behind  the  E.  M.  F.,  and  consequently  with  the  self- 
induction  of  the  circuit.  The  calorific  power  is  zero  for 

0  =  —  ,  i.e.,  for  a  lag  of  \  of  a  period. 
In  the  expression 

p     -1      *•' 
2r   i   a*L* 
r 

if  Ls  is  constant,   the  denominator  is  minimum  for 


consequently  the  value  of  the  resistance  of  the  circuit  which 
renders  the  mean  calorific  power  a  maximum  is 

r  =  aLs. 
As  then 


we  have 


and  the  lag  corresponding  to  the  maximum  power  is  equal 
to  £  period.     This  power  is  expressed  by 


APPLICATIONS   OF   THE   LAWS   OF  INDUCTION.     305 

201.  Mutual  Induction  of  Two  Circuits.  —  Suppose  that 
two  circuits  of  invariable  form,  having  resistances  r  and  r' 
and  a  coefficient  of  mutual  induction  equal  to  Lm  (§  143), 
approach  each  other  so  slowly,  that  the  currents  z,  i'y  which 
traverse  them,  can  be  considerecLas  constant. 

For  an  elementary  displacement,  denoting  by  E,  E'  the 
electromotive  forces  of  the  sources  producing  the  currents, 
we  have 


dt       ., 

»=  —  -  —  •'= 


From  these  equations  we  get 

(Ei  +  E'i'}dt  -  (i*r  -f  *'V)d/  =  2u'dLm. 

(Ei  +  E  fi')dt  expresses  the  energy  furnished  by  the  gen- 
erators during  a  time  dt  ;  (fr  -{-  i'^r')dt  is  the  portion  of  this 
energy  which  is  transformed  into  heat  in  the  conductors. 

oy 

ii'dLm  is  the  work  of  the  electrodynamic  forces.  As  the 
excess  of  the  energy  expended  over  that  transformed  into 
heat  is  double  this  work,  we  conclude  that  a  portion  equal 
to  ii'dLm  is  stored  up  in  the  system  in  the  state  of  potential, 
or  intrinsic,  energy. 

It  will  be  remembered,  in  fact  (§  143),  that  the  mutual 
energy  of  two  circuits  is  expressed  by  —  ii'Lm  ;  its  variation 
is  therefore,  of  course,  ii'dLm. 

202.  Mutual  Induction  of  Two  Fixed  Circuits.—  Two 

circuits,  invariable  in  form  and  position,  have  resistances  r 
and  r'  ,  coefficients  of  self-inductance  Ls  and  Z-/,  and  a  coeffi- 
cient of  mutual  inductance  Lm  (§  143). 

If  cells  having  electromotive  forces  E  and  E'  are  in  the 
circuits,  the  currents  at  any  given  instant  of  the  variable 
period  will  be 


ELECTROMAGNETIC   INDUCTION. 


i=  —        —  —,       .    .    .    .    (I) 


(2) 


As  the  circuits  are  fixed,  Lm,  Ls  and  LJ  are  constants,  so 
that  on  multiplying  (i)  and  (2)  respectively  by  idt  and  i'dt 
and  then  adding,  we  get 

(Ei+E'i')dt  -  (i*r  +  i'*r')dt 

=  L,idt  +  L.'t'dt'  +  Lm(idi'  +  i'di). 

We  see  that,  in  this  case,  the  excess  of  energy  furnished 
by  the  cells  over  the  energy  transformed  into  heat  is 


Ltidi  +  Ls'ifdi'  +  LJJdi'  +  i'di). 
This  expression  is  the  exact  differential  of 


which  represents  the  potential  energy  of  the  circuits  when 
the  currents  have  values  z',  i'  '. 

The  first  two  terms  represent  the  intrinsic  energy  of  each 
circuit,  and  the  third  is  their  mutual  energy. 

203.  Quantity  of  Induced  Electricity.  —  Let  us  take  the 
case  where  E'  =  o;  the  current  of  the  second  circuit  is  then 
alone  the  cause  of  the  mutual  induction,  and  equation  (2) 
gives  by  integration 


for  the  induced  current  is  zero  at  the  beginning  and  end  of 
the  variable  period  of  the  inducing  current. 


APPLICATIONS  OF  THE  LAWS   OF  INDUCTION,     3O/ 
Consequently 

A-M/    *'-     L™i-    L-E 
J,  *&-<?>  -  --?-1-    --fr- 

\ 
When  the  inducing  circuit  is  broken,  we  have  likewise 


The  quantities  of  electricity  induced  are  equal  and  of 
opposite  sign  in  the  two  cases. 

204.  Expression  for  Mutual  Inductance.—  We  have 
shown  (§  163)  that  the  sum  of  the  magnetic  fluxes  which 
traverse  the  coils  of  a  cored  annular  bobbin,  divided  by 
the  current  in  the  coils,  is  expressed  by 

L,  =  ^Ttn^^s  =  ^7tn*}Jilst 

when  the  diameter  of  the  ring  is  very  great  compared  to  its 
thickness.  If  we  suppose  that  the  first  bobbin  is  entirely 
covered  by  a  second,  having  n'  turns  per  unit  of  length 
along  the  axis,  and  n'  total  turns,  the  coefficient  of  induc- 
tion of  this  second  bobbin  is 

Lt'  =  47rnlfnr^is  =  qnn^pls. 

Now  the  coefficient  of  mutual  induction  or  the  mutual 
inductance  is  the  ratio  of  the  flux  produced  by  one  of  the 
bobbins  across  the  coils  of  the  other,  to  the  current  which 
traverses  the  first. 

Lm  —  ^nn^s  X  ri  =  ^nn{  f*s  X  n  — 
We  see  therefore  that  we  then  have  the  relation 


3O8  ELECTROMAGNETIC  INDUCTION. 

This  simple  expression  is  applicable  whenever  the  lines 
of  force  generated  by  one  of  the  bobbins  traverse  all  the 
turns  of  the  other.  It  is  the  maximum  value,  therefore,  of 
the  mutual  induction  of  the  two  circuits.  This  condition  is 
realized  in  the  middle  region  of  two  concentric  solenoids  of 
great  length  and  with  a  rectilinear  axis.* 

205.  Induction  in  Metallic  Masses.— In  what  has  pre 
ceded,  we  have  had  particularly  in  view  the  phenomena  of 
induction  developed  in  linear  circuits  ;  but  it  is  evident  thai- 
induction  takes  place  in  metallic  masses  of  any  shape  when 
they  cut  lines  of  force  in  a  magnetic  field. 

Faraday's  disc  (§  191)  shows  the  development  of  induced 
currents  in  the  case  of  a  solid  disc  turning  between  the 
poles  of  an  electromagnet.  The  determination  of  the  lines 
of  electric  flux  or  current  in  such  a  case  presents  great 
complexity. 

To  resolve  the  problem,  we  put  against  the  disc  copper 
points  connected  with  a  galvanometer,  proceeding  as  shown 
in  §  168.  In  this  way  we  get  series  of  points  at  the  same 
potential,  which  enables  us  to  draw  equipotential  lines.  The 
lines  of  flux  are  perpendicular  to  these. 

If  Faraday's  disc  is  made  to  revolve  without  connecting 
the  sliding  contacts  by  a  conductor,  the  lines  of  current  be- 
come closed  on  themselves  in  the  disc,  producing  curves 
which  envelop  each  other  without  intersecting,  and  which 
are  divided  into  two  separate  groups  by  a  vertical  plane 
passing  through  the  axis  of  rotation. 

By  Lenz'  law  the  direction  of  these  currents  is  such  that 
they  tend  to  oppose  the  movement  of  the  disc  ;  consequently 
the  currents  which  approach  the  poles  are  in  the  opposite  di- 
rection to  the  currents  of  a  solenoid  equivalent  to  the  induc- 

*  See  Mascart  and  Joubert,  Lemons  sur  FElectridtt  et  le  Magnetisme  for  the 
working  out  of  calculations  on  inductances  of  solenoids. 


APPLICATIONS   OF   THE   LAWS   OF  INDUCTION.     309 

ing  magnet.    The  currents  which  move  away  from  the  poles 
are  in  the  same  direction'  as  would  be  the  solenoidal  ones. 


206.  Foucault  Currents. — jThis  consequence  of  Lenz* 
law  is  shown  in  various  ways ;  t^e  following  experiment  is 
due  to  Foucault.  A  rapid  movement  of  rotation  is  given 
to  a  disc  embraced  by  the  pole-pieces  of  a  powerful  electro- 
magnet. At  the  moment  when  a  current  is  passed  through 
the  latter,  the  mechanical  resistance  caused  by  the  induced 
currents  causes  the  stoppage  of  the  disc  ;  if  the  motion  is 
continued  by  a  sufficient  expenditure  of  motive  power,  the 
disc  grows  hot  in  consequence  of  the  Joule  effect. 

The  induced  currents  are  very  greatly  reduced,  and 
consequently  the  resistance  to  rotation  and  the  heating,  by 
dividing  the  disc  by  slits  normal  to  the  direction  of  the  E. 
M.  Fs.  of  induction  and  thus  cutting  the  lines  of  flux.  In 
the  present  case  these  divisions  would  be  circles  concentric 
with  the  disc,  and  this  latter  would  have  to  be  made  of  rings 
of  increasing  diameter,  separated  by  an  insulating  material. 

The  currents  induced  in  metallic  masses  are  usually  com- 
prised under  the  name  of  Foucault  or  eddy  currents. 

The  mechanical  resistance  caused  by  the  induced  currents 
in  these  masses  is  utilized  to  deaden  the  movement  of 
galvanometer-needles. 

If,  for  example,  we  surround  .a  magnet,  movable  around 
an  axis  of  suspension,  with  a  mass  of  copper  in  which  is 
hollowed  a  cavity  sufficient  to  permit  the  swings  of  the 
magnet,  the  latter  will  develop  in  the  copper  induced  cur- 
rents which  oppose  its  movement  and  cause  its  stoppage. 

When  a  conductive  rod  which  forms  part  of  a  circuit  is 
displaced  across  the  lines  of  force  of  afield  presenting  varia- 
tions of  intensity  at  different  points,  besides  the  E.  M.  F.  of 
induction  observed  in  the  circuit,  Foucault  currents  are 
generated  in  the  substance  of  the  conductor.  In  fact  the 


310  ELECTROMAGNETIC  INDUCTION. 

elementary  filaments  which  constitute  the  conductor  cut  at 
the  same  instant  different  numbers  of  lines  of  force.  These 
subsidiary  currents  heat  the  rod  without  doing  useful  work 
in  the  circuit. 

207.  Cores  of  Electromagnets  Traversed  by  Variable 
Currents.  Calculation  of  the  Power  Lost  in  Foucault 
Currents. — Foucault  currents  tend  to  be  set  up  in  the  core 
of  an  electromagnet  whose  coils  are  traversed  by  a  periodic 
current.  To  diminish  these  currents,  which  heat  the  iron 
uselessly,  the  core  is  made  of  thin  plates  insulated  from 
each  other  by  varnish,  or  varnished  or  paraffined  paper,  and 
put  together  in  such  a  way  that  their  surfaces  of  separation 
are  parallel  to  the  axis  of  the  coils  and  consequently  cut  the 
directions  of  the  E.  M.  Fs.  of  induction.  The  bolts  used 
to  fasten  the  plates  together  should  be  insulated  by  tubes 
of  vulcanized  fibre  and  washers  of  the  same  put  under  the 
heads  of  the  bolts  and  the  nuts.  It  is  a  good  plan  to  wrap 
the  whole  core  round  with  varnished  cloth,  so  that  the 
edges  of  the  plates  may  not  pierce  the  insulation  of  the 
wires  rolled  on  the  iron. 

A  core  is  sometimes  made  of  a  bundle  of  varnished  iron 
wires,  but  it  is  to  be  remarked  that  the  space  lost  by  the 
interstices  between  the  wires  is  much  greater  than  in  the 
case  of  a  laminated  core.  Moreover  this  arrangement  is 
only  applicable  to  small,  straight  electromagnets.  As  the 
wires  do  not  touch  except  along  single  lines,  the  insulation 
need  not  be  so  careful  as  that  of  the  plates.  The  division 
of  the  core  parallel  to  its  axis  enables  the  heating  by  Fou- 
cault currents  to  be  avoided,  but  not  that  which  results 
from  hysteresis  (§  6 1 ).  Under  the  action  of  the  periodic 
current  of  the  coils,  the  core  is,  practically,  subjected  to 
successive  magnetizations  in  opposite  directions  which  cause 
a  loss  of  energy  in  proportion  to  the  variations  of  the  mag- 
netizing force  and  the  coercive  force  of  the  core. 


APPLICATIONS   OF   THE  LAWS   OF  INDUCTION.     311 

It  is  possible  to  calculate  the  loss  occasioned  by  Foucault 
currents  in  an  iron  plate  or  wire  taken  in  the  core  of  an 
electromagnet  traversed  by  periodical  currents. 

Take  the  case  of  a  cylindrical  wire  of  length  /  and  radius 
R,  traversed  longitudinally  by  ^variable  flux  which  tends  to 
develop  eddy  currents  parallel  to  the  edge  of  a  right  section 
of  the  wire.  Let  us  take  inside  the  wire  a  concentric  tube 
infinitely  thin,  of  radius  r  and  thickness  dr.  Every  variation 
of  the  flux  along  the  wire  sets  up  in  the  tube  an  E.  M.  F. 
of  induction  expressed  by 


The  resistance  opposed  to  the  current  by  the  tube  is 

2nrp 


p  being  the  specific  resistance  of  the  metal. 

The  power  set  free  in  the  form  of  heat  in  the  tube  is  the 
ratio  of  the  square  of  the  E.  M.  F.  to  the  resistance,  or 


v~      d  t          ,  /  /day, 

dp  =  -    -  =  nr*  —  l-r-J  dr. 
27i  rp  2p\dt  I 

~~ldr~ 
The  total  loss  in  the  wire  is,  consequently, 


r*    ,  /  /day,       m?t/d<a\* 
t  =       ^WJdr  =  Tyns 

*/0 

The  loss  of  power  per  cm.s  is 


312  ELECTROMAGNETIC  INDUCTION. 

In  the  case  of  a  plate  of  length  /,  thickness  n,  and  breadth 
m,  we  would  have  found,  supposing  the  lines  of  current 
parallel  to  the  edges  of  the  plate  (which  is  only  approxi- 
mately exact), 

_p_       n*  i 

Imn  ~  6p  \ 

In  each  case  we  observe  that  the  loss  diminishes  rapidly 
with  the  thickness  of  the  elements  of  the  core. 

From  the  researches  of  Messrs.  J.  Thomson  and  Ewing* 
it  appears  that  the  Foucault  currents,  produced  in  the  ele- 
ments of  the  core  of  an  electromagnet  traversed  by  alter- 
nating currents,  exercise  a  magnetizing  effect  on  the 
molecules  of  iron  inside  the  plates  or  wires  composing  the 
core,  which  effect  is  the  opposite  of  that  exercised  by  the 
current  in  the  field  coils.  It  follows  that  in  order  to  obtain  a 
given  total  magnetic  flux,  the  magnetic  induction  in  the 
outer  layers  of  the  wires  or  plates  must  be  forced  up,  which 
causes  an  increase  in  the  loss  by  hysteresis,  since  this  latter 
increases  more  rapidly  than  the  maximum  induction  to 
which  the  iron  is  subjected. 

208.  Self-induction  in  the  Mass  of  a  Cylindrical  Con- 
ductor. Expression  for  the  Coefficient  of  Self-induction 
in  such  a  Conductor. — When  a  variable  current  flows  in  a 
cylindrical  conductor,  the  current-density  is  not  constant 
over  the  whole  section  of  the  conductor ;  it  is  greater  at  the 
periphery  than  at  the  centre.  To  account  for  this  fact  we 
can  mentally  divide  the  total  current  into  an  infinite  num- 
ber of  elementary  parallel  currents,  capable  of  reacting 
upon  each  other.  The  current  which  passes  in  one  filament 
tends  to  induce  inverse  currents  in  the  neighboring  fila- 
ments. These  mutual  reactions  are  all  the  stronger  since 

*  Electrician,  April  22,  1892. 


APPLICAJ^IONS   OF   THE  LAWS   OF  INDUCTION.      313 

the  filaments  are  crowded  together.  A  reduction  of  the 
current  therefore  follows,  which  is  a  maximum  towards  the 
middle  of  the  conductor's  section,  and  minimum  at  the 
periphery.  The  conductor  presents  for  variable  currents 
a  resistance  which  is  higher  th&n  that  with  constant  cur- 
rents. 

The  induction  in  the  mass  of  the  conductor  is  further  in- 
creased in  the  case  of  a  magnetic  wire,  such  as  an  iron  one, 
in  consequence  of  the  circular  magnetization  which  the 
metal  then  assumes,  and  which  is  due  to  the  circular  lines 
of  force  generated  by  the  filaments  of  the  current  inside 
(§  135).  As  was  shown,  however,  in  §  188,  this  effect  does 
not  appear  except  in  the  case  of  alternating  currents ;  with 
intermittent  currents,  constant  in  direction,  the  molecules 
of  the  iron  kept  their  orientation  in  the  form  of  closed 
chains  and  do  not  produce  any  induction  effect ;  with  alter- 
nating currents  it  is  preferable  to  use  conductors  of  some 
non-magnetic  metal. 

When  the  period  of  variation  is  excessively  short,  as  is 
the  case  in  discharging  a  condenser,  it  may  be  conceived 
that  the  mutual  reactions  carry  the  whole  current  towards 
the  external  layers  of  the  wire.  In  proportion  as  the 
period  of  the  current  decreases,  we  are  thus  led  to  separat- 
ing the  elementary  filaments  more  and  more  and  to  giving 
the  conductors  the  greatest  surface  possible.  Metallic 
ribbons,  tubes,  and  cords  are  then  preferable  to  conductors 
of  circular  section. 

To  arrive  at  the  value  of  the  coefficient  of  self-induction 
of  a  cylindrical  conductor,  let  us  first  find  that  of  a  circuit 
formed  of  two  conductors  G,  Gf,  parallel  and  long  enough 
to  be  considered  as  indefinite.  Such  would  be  the  case  with 
two  neighboring  telegraph-wires.  Let  /  be  the  current 
traversing  the  circuit,  r  the  radius  of  the  conductors,  and  d 
the  distance  between  their  axes. 


314  ELECTROMAGNETIC  INDUCTION. 

Let  us  determine  the  flux  of  force  produced  by  the  con- 
ductors in  the  space  limited  by  their  axes  and  by  two  planes 
normal  to  them,  I  cm.  apart.  Each  of  the  conductors 
evidently  produces  half  the  flux.  Denote  by  /*  the  per- 
meability of  the  surrounding  medium,  and  by  /*'  that  of 
the  wires. 

The  conductor  G  produces  in  an  external  point,  situated 
at  a  distance  a  from  its  axis,  a  field  whose  intensity  is  the 
same  as  if  the  current  were  condensed  along  the  axis,  or 

2,i 

—  (§  146).     The  corresponding  magnetic  induction  is  conse- 

2  Ut 

quently  -£- .  If  we  imagine  in  the  above-defined  space  a 
section  parallel  to  G,  of  thickness  da,  the  flux  across  this 
elementary  surface  is •  — .  The  total  flux,  due  to  G, 

which  cuts  the  surface  comprised  between  the  edge  of  G  and 
the  axis  of  G'  is  therefore  per  unit  length 


I 


td2uida  d 

=  2/«  log,  -. 


In  the  part  of  the  space  under  consideration  occupied 
by  G,  the  field  has  a  different  expression.  In  a  point  taken 
in  the  interior  of  G  at  a  distance  b  from  the  axis,  the  field 
is  the  same  as  that  which  would  be  produced  by  a  current 
condensed  along  the  axis  and  bearing  the  same  proportion 
to  the  total  current  as  a  cross-section  of  radius  b  bears  to 
the  total  cross-section  of  the  conductor. 

We  will  then  have  for  the  field-intensity 

2Z  7t&  2lb 

J  X  ^  z  :  7s"' 
and  for  the  value  of  the  magnetic  induction  at  the  point  b 


APPLICATIONS   OF   THE  LAWS   OF  INDUCTION.     315 

The  flux  traversing  half  the  longitudinal  section  of  G, 
and  which  traverses  the  mass  of  the  conductor,  will  be,  per 
unit  length  along  the  axis, 


r 

The  total  flux  due  to  G  is  therefore 

it 


As  £'  furnishes  an  identical  flux  across  the  space  under 
consideration,  we  will  get  altogether 


21  (2/1  log,-  +/). 


By  definition,  the  self-inductance  of  the  circuit  will  there- 
fore be,  per  unit  of  length, 


L.t  =  22,1  log.     +  x  ......     (i) 

In  the  case  of  copper  conductors  suspended  in  air  we 
have  practically  /*  =  //  =  I,  whence 

£'.,=  2(2168,-  +  l}..      .      .      ...      .      (2) 

It  will  be  noted  that  the  logarithmic  expression,  giving 
the  value  of  the  flux  in  the  space  between  the  two  conduc- 
tors, shows  that  the  induction  decreases  very  rapidly  as  we 
get  further  from  these  latter  ;  that  is,  beyond  a  certain  dis- 
tance between  the  conductors,  the  flux  is  not  sensibly 
increased  by  augmenting  that  distance. 

We  can  therefore  say  that  in  a  circuit  of  any  form  what- 
ever, the  self-inductance  is  proportional  to  the  length  of  the 


3l6  ELECTROMAGNETIC  INDUCTION. 

conductors  composing  the  circuit,  provided  that  these  con- 
ductors be  sufficiently  far  apart. 

The  part  of  the  coefficient  due  to  the  sectional  dimensions 
of  each  conductor  is  simply  expressed  by  X°  This  quantity 
is  negligible  when,  JJL  being  equal  to  //,  the  distance  d  is 
great  compared  to  r,  or  else  when  the  permeability  /*  is  very 
great  compared  to  //.  We  have  an  example  of  the  former 
in  the  case  of  two  copper  wires  stretched  in  the  air,  like 
telegraph-wires.  The  second  case  occurs  in  a  copper  con- 
ductor wound  round  an  iron  core  of  large  diameter. 

The  above  reasoning  shows  that  the  flux  traversing  the 
mass  of  conductors  increases  from  their  axis  to  their 
periphery.  This  way  of  reasoning  takes  for  granted  that 
the  initial  spreading  of  the  current  is  uniform  throughout 
the  conductor's  cross-section.  If  the  current  is  variable, 
the  effect  of  the  unequal  spreading  of  the  flux  over  the 
cross-section  is  to  create  E.  M.  Fs.  of  induction  which  tend 
to  increase  the  density  of  the  current  towards  the  periphery 
and,  consequently,  to  still  further  increase  the  flux  in  this 
region.  The  consequence  of  these  reactions  is  an  increase 
of  resistance ;  and  if  the  frequency  of  the  current  is  suffi- 
ciently great,  the  surface  layers  of  the  conductor  alone  are 
concerned  in  the  electric  displacement. 

This  effect  is  very  clearly  shown  when  a  condenser  is  re- 
peatedly discharged  into  a  thick  copper  wire :  an  incandes- 
cent lamp  can  be  made  to  glow,  when  put  in  shunt  around 
a  very  small  length  of  the  wire,  by  reason  of  the  difference 
of  potential  caused  by  the  resistance  of  the  external  layers 
of  the  conductor,  which  are  the  only  ones  affected.  Under 
these  circumstances  only  the  surface  of  the  wire  is  heated 
by  the  current,  the  interior  being  heated  by  conduction. 
This  can  be  shown  by  causing  a  tolerably  large  current  of 
high  frequency  to  pass  through  a  wire  for  a  very  brief  time  : 
the  wire,  after  growing  hot  for  a  moment  on  its  surface, 


APPLICATIONS   OF   THE   LAWS   OF  INDUCTION.     317 

rapidly  grows  cool  by  diffusion  of  the  heat  throughout  its 
mass. 

M.  Potier  has  given  the  following  formula  to  express  the 
ratio  between  the  resistance  R+  6f  a  round  wire  for  alternat- 
ing currents  of  period  T,  and  th^  resistance  Rc  of  the  same 
wire  for  continuous  currents.  Denoting  by  /  the  length  of 
the  wire,  d  its  diameter,  and  p  its  specific  resistance,  we  have 


R'  =  P~» 


-j  . -_     _, 

Rc  ~        *~  '  4 


and 


With  a  frequency  of  80,  the  increase  of  resistance  is  2.5 
per  cent,  for  a  wire  15  mm.  in  diameter;  it  is  8  per  cent,  at 
a  frequency  of  133.  The  resistance  Ra  calculated  by  the 
above  formula  is  that  which,  multiplied  by  the  square  of 
the  effective  current,  gives  the  power  lost  in  the  Joule  effect. 
It  will  be  seen  that,  in  order  to  avoid  excessive  cost  of  plant 
and  working,  conductors  for  strong  alternating  currents 
must  have  the  form  of  tubes. 

ROTATIONS    UNDER  THE   ACTION   OF   INDUCED    CURRENTS. 

20p.  The  reaction  of  the  induced  currents,  in  accordance 
with  Lenz'  law,  enables  us  to  obtain  various  movements  of 
rotation. 

Arago  had  noticed  that  if  a  copper  disc  rotates  underneath 
a  magnetized  needle  placed  on  a  pivot  over  the  centre  of 
the  disc,  the  needle  is  drawn  in  the  direction  of  the  disc's 
movement. 

Likewise,  if  a  strong  magnet  be  rotated  above  a  movable 
metallic  disc,  the  latter  tends  to  take  up  the  same  motion 
as  the  magnet.  Referring  to  the  explanation  given  in  §205, 
we  see  that  the  currents  induced  in  front  of  the  poles  are 


ELECTROMA  GNE TIC  2ND  UCTION. 

repelled  by  the  solenoidal  currents  equivalent  to  the  mag- 
net. The  currents  induced  behind  the  poles  are,  on  the 
contrary,  attracted. 

This  experiment  enables  us  to  formulate  a  general  rule: 
Whenever  the  lines  of  force  of  a  magnetic  field  are  dis- 
placed in  a  conductor,  it  tends  to  follow  the  movement  of 
the  field. 

210.  Ferraris'  Arrangement. — We  are  indebted  to 
Ferraris  for  some  ingenious  arrangements  based  on  the 
application  of  this  rule. 

Suppose  a  conductive  disc  movable  about  its  axis  O  and 


enclosed  between  two  pairs  of  coils  AA ',  BB',  shown  in 
section  in  the  figure. 

Similar  periodic  currents  are  passed  through  these  coils, 
but  differing  in  phase  by  90°. 

A  part  of  the  lines  of  force  thus  produced  traverses  the 
substance  of  the  disc,  entering  by  the  edge.  Denote  by  3C 
the  mean  intensity  of  the  field  formed,  at  any  moment,  in 
the  disc,  by  the  coils  A  A'  and  directed  along  the  axis  of 
these  coils.  3Cm  being  the  maximum  intensity,  we  get 

OC  —  OCm  sin  at. 
If  we  suppose  that  the  coils  BB'  produce  a  field  whose 


APPLICATIONS  OF  THE  LAWS   OF  INDUCTION.     319 

maximum  intensity  is  also  3Cm,  but  lagging  a  quarter-period 
behind,  its  mean  intensity  directed  normal  to  OC  is 

3C'  =  3Cm  sin  f  at  —  »— 1  =  —  3Cm  cos  at. 

^ 

The  two   fields  are  compounded   in   the  direction   of  a 
resultant  which  has  at  each  instant  an  intensity 


je"  =  V2oc  -hoe /a  =  ocm. 

We  see  that  the  resultant  intensity  is  constant  in  magni- 
tude, but  that  its  direction  varies.  For  at  —  o,  VI"  has  the 

same  direction  as  3C' ;  when  at  increases  from  o  to  — ,  the 
resultant  field  turns  about  O  and  becomes  coincident  with 
3C,  For  values  of  at  comprised  between  -  and  n  the  direc- 
tion of  the  resultant  performs  another  quarter-revolution  ; 
it  returns  to  its  initial  position  for  at  =  2n. 

To  sum  up,  the  magnetic  field  produced  by  the  combined 
action  of  the  two  pairs  of  coils  revolves  about  O.  The  con- 
ductive disc  is  then  traversed  by  induced  currents  whose 
reaction  urges  it  to  turn  in  the  same  direction  as  the  field. 

If  the  two  pairs  of  coils  were  traversed  by  currents  of  the 
same  phase,  the  direction  of  the  resultant  field  would  remain 
constant  in  the  plane  which  bisects  all  the  coils,  but  the 
intensity  would  vary  from 

+  OCm|/2     to      -JC^VI 

For  every  difference  of  phase  between  o°  and  90°,  a  rotat- 
ing field  will  be  obtained  ;  the  extremity  of  the  straight 
line  representing  the  resultant  will  describe,  not  a  circum- 
ference as  in  the  first  case,  but  an  ellipse  whose  major  axis 
will  lie  along  the  bisecting  plane. 


320  ELECTROMAGNETIC  INDUCTION, 

211.  Shallenberger's  Arrangement.  —  Shallenberger  has 
invented  an  arrangement  which  enables  an  analogous  rota- 
tion to  be  obtained  by  the  aid  of  a  single  periodic  current, 
and  this  he  has  applied  in  the  design  of  an  alternating-cur- 
rent meter. 

A  coil  of  oblong  form,  traversed  by  a  periodic  current, 
surrounds  a  second  smaller  coil  whose  axis  is  inclined  at 
about  45°  to  that  of  the  larger  coil.  If  we  denote  by  Lm 
their  mutual  inductance,  the  E.  M.  F.  induced  in  the  second 
coil  is 


This  expression  shows  that  the  induced  E.  M.  F.  is  J 
phase  behind  the  inducing  current.  In  consequence  of  self- 
induction,  the  induced  current  likewise  lags  behind  the 
E.  M.F.,  so  that  the  phases  of  the  inducing  and  the  induced 
currents  differ  by  an  angle  comprised  between  90°  and  180°. 
Hence  it  follows  that  inside  the  coils  is  the  seat  of  a  rotat- 
ting  magnetic  field,  which  carries  with  it  in  its  rotation  a 
disc  movable  about  a  vertical  axis.  The  effect  is  increased 
by  using  a  disc  of  iron  whose  permeability  increases  the  in- 
tensity of  the  field. 

212.  Repulsion  Exercised  by  an  Inducing  Current  on 
an  Induced  Current.  —  If  we  represent,  by  the  graphic 
method  of  §  199,  two  neighboring  currents  lagging  one  be- 
hind the  other  by  a  quarter  period,  it  is  easy  to  see  that 
during  one  half  of  their  phase  they  are  in  the  same  direction 
and  attract  each  other  ;  during  the  other  half,  they  are  in 
opposite  directions  and  repel  each  other:  the  resultant  of 
the  attractive  forces  is,  moreover,  equal  to  that  of  the  repul- 
sive forces,  so  that  their  mutual  action  may  give  rise  to  vi- 
brations, but  there  will  be  no  resultant  effort  of  translation. 

We  have  just  seen  that  the  inducing  current  is   always 


APPLICATIONS   OF   THE   LAWS   OF  INDUCTION.     $21 

more  than  90°  ahead  of  the  secondary  current  on  account 
of  the  time-constant  of  the  secondary  circuit :  such  is  the 
case  for  the  currents  shown  by  full  lines  in  Fig.  95. 

The  repellant  period  during"  which  the  currents  are  in 
opposite  directions  has  then  a£ greater  duration  than  the 
period  of  attraction  ;  moreover  the  figure  shows  that  during 


FIG.  95. 

the  period  of  repulsion  the  mean  currents  are  greater  than 
during  the  attraction.  It  follows  that  a  periodic  inducing 
current  repels  the  induced  current  with  a  force  which  is 
greater  as  the  self-induction  of  the  secondary  circuit  be- 
comes greater. 

Prof.  Elihu  Thomson  has  based  some  interesting  experi- 
ments on  these  observations.  If,  for  example,  we  put  a 
metallic  ring  A  around  the  upper  end  of  a  straight  electro- 
magnet B  traversed  by  alternating  currents,  it  is  found  that 
the  ring  is  repelled  to  a  position  A'.  On  interposing 
a  metallic  plate  between  the  magnet  and  the  ring,  the  ring 
is  observed  to  be  no  longer  repelled.  This  is  because  the 
screen  itself  becomes  the  seat  of  an  induced  current  whose 
effect  on  the  ring  neutralizes  that  of  the  primary  current. 
We  shall  see  in  the  following  chapter  that  such  a  screen 


322  ELECTROMAGNETIC  INDUCTION    - 

intercepts  the  electromagnetic  waves  through  the  action  of 
which  induction-effects  are  produced. 

When  we  consider  the  repulsive  effects  with  regard  to 
the  inducing  periodic  magnetic  field,  we  observe  that  the 
circuit  which  is  repelled  tends  to  place  itself  so  that  the 
periodic  flux  traversing  it  shall  be  a  minimum,  for  then  the 


FIG.  96. 

induced  currents  which  cause  the  movement  are  reduced  as 
much  as  possible. 

The  curious  arrangements  made  by  Ferraris  and  Prof. 
Elihu  Thomson  contain  the  germ  of  alternating-current 
motors. 

Prof.  Fleming  has  made  an  apparatus  for  measuring  pe- 
riodic currents,  in  which  use  is  made  of  the  phenomena  of 
repulsion  above  described.  A  copper  disc  D  (Fig.  97)  is 
hung  at  the  centre  of  a  coil  C>  traversed  by  the  undulatory 
currents. 

The  disc,  being  placed  at  aa'  at  an  angle  of  45°  with  the 
coil,  is  the  seat  of  induced  currents,  and,  in  consequence  of 
the  repulsion  exercised  by  the  primary  current,  tends  to 
place  itself  in  the  position  bb' ,  for  which  the  periodic  flux 
traversing  it  is  a  minimum.  The  mirror  M  enables  us  to 
read  by  reflection  the  angle  at  which  the  disc  comes  to  rest, 
under  the  influence  of  the  electrodynamic  forces  on  one 
hand  and  the  torsion  of  the  suspension-wire  on  the  other. 


APPLICATIONS   OF  THE  LAWS  OF  INDUCTION.     323 

The  disc  must  be  placed  obliquely  to  the  coil,  for  if  it 
were  parallel  to  the  turns  of  the  coil  it  would  oscillate  con- 


FIG.  97. 

tinuously,  the  current  giving  it  a  new  impulse  at  each  cross- 
ing of  the  plane  normal  to  bbf. 


IMPEDANCE* 

213.  Inductance. — A  sinusoidal  E.  M.  F.  may  be  ex- 
pressed by 

e  =  E  sin  cot, 

where     e  is  the  instantaneous  E.  M.  F.  in  volts  ; 

E  is  the  maximum  cyclic  E.  M.  F.  in  volts  ; 
GO  is  the  angular  velocity  of  the  E.  M.  F.  in  radians 
per  second,  and  is  2nn,  where  n  is  the  numb.er 
of  cycles  of  the  E.  M.  F.  per  second  ; 
t  is  the  time  expressed   in  seconds,   starting  from 
some   suitable    instant    at  which   E   is  zero   and 
changing  from  negative  to  positive. 

When  such  an  E.  M.  F.  is  impressed  upon  a  circuit  hav- 
ing a  resistance  of  R  ohms  and  an  inductance  of  L  henrys, 
a  certain  current  strength  will  pass  through  the  circuit. 
We  assume  that  the  inductance  is  non-ferric,  that  is  to  say, 
that  the  circuit  is  not  linked  with,  or  situated  in  the  neigh- 
borhood of,  iron,  so  that  if  there  are  coils  of  wire  in  the 
circuit,  these  coils  are  without  cores.  Then  the  current 
in  the  circuit  will  also  be  sinusoidal.  This  may  be  shown 
as  follows: 

The  counter  E.  M.  F.  (C.  E.  M.  F.)  in  the  circuit  at  any 

instant  due  to  the  action  of  the  inductance  will  be  —  L-r 

at 

volts,  where  i  is  the  instantaneous  current  strength  in  am- 
peres.    The  apparent  C.  E.  M.  F.  due  to  fall  of  potential 

*  By  A.  E.  Kennelly. 

324 


IMPEDANCE.  325 

in  the  ohmic  resistances  will  be  —  iR  volts;  and  if  no  elec- 
trostatic capacity  exists  in  the  circuit,  these  will  be  the 
only  sources  of  C.  E.  M.  F.  present.  But  the  impressed 
E.  M.  F.,  e,  must  be  equal  and  opposite  at  any  and  every 
instant  to  the  tota.1  C.  E.  M. 


e  =•  E  sin  cot  =  iR  +  L-j-  volts. 

The  solution  of  this  differential  equation  within  a  con- 
stant  which  is  negligible  as  soon  as  the  current  has  become 
steady,  is 


E  sinf  oot  —  tan 
*  = — 


If  the  circuit  contains  resistance  only,  or  L  =.  o,  this  be- 
comes 

£  sin  cot       e  t  ' 

t  =^     — -j) —  ^>  amperes 0    .    (2) 

That  is,  the  current  i  would  be  in  phase  with  the  E.  M.  F. 

and  would  have  at  each  instant  the  value  -5.     The  intro- 

J\ 

duction  of  the  inductance  into  the  circuit  has  altered  both 
the  magnitude  of  the  current  and  its  phase.  The  magni- 
tude becomes 


which  is  always  less  than  ^,  while   the   phase  has  become 
retarded  by  the  angle  tan"1-^-,  which  is  always  less   than 


326  IMPEDANCE, 

This  effect  of  inductance  may  be  considered  from  two 
distinct  points  of  view,  namely  : 

1st,  and  fundamentally,  we  may  regard  the  circuit  as 
having  simply  a  resistance  R,  but  as  being  the  seat  of  two 
separate  although  interdependent  E  .M.  Fs.,  one  being  the 
impressed  sinusoidal  E.  M.  F.,  e  volts,  and  the  other  being 

the  self-induced  E.  M.  F.,  L~  volts. 

at 

Since         i  =  /  sin  oot  amperes, (3) 

—  —  IGD  cos  Got  =  loo  sin  f  oat  -| —  ) ,     .     .     .     (4) 

/.  L-r  =  ILdD  sin  f  oat  -\ — j  volts, (5) 

This  shows  that  the  self-induced  E.  M.  F.  has  the  maxi- 
mum value  ILoo  volts,  that  the  effective  value  (square  root  of 

mean  square)  is  -~=Lo)  volts,  and  that  the  phase  of  this  E. 


M 


.  F.  is  90°  is  rr  1  ahead  of  the  current  /. 


If,  therefore,  we  represent  the  counter  E.  M.  F.  due  to 
the  fall  of  potential  (iR  volts)  at  any  instant,  say  for  con- 
venience at  its  maximum  value,  by  the  straight  line  01, 


Lie) 


FIG.  97. 
Fig.  97,  then  the  product  of  this  value  into  —  will  give  the 

induced  E.  M.  F.  when  turned  through  a  right  angle,  or  di- 
rected as  shown  by  the  line  IE. 


IMPED  A  NCR.  327 

The  geometrical  sum  of  these  two  E.  M.  Fs.  will  be  OE 
in  both  direction  and  magnitude,  and  this  will  be  the  im- 
pressed E.  M.  F.,  e,  at  the  instant  considered. 

According  to  this  conceptiqn;*  the  E.  M.  F.  iR  which  acts 
upon  the  resistance  R  so  as  tq^orce  through  it  the  current 
t,  is  the  geometrical  difference  OI  of  the  impressed  E.  M.  F. 
OE  and  the  self-induced  E.  M.  F.  IE.  In  order  to  deter- 
mine the  current  strength  z,  we  have  to  compute  the  influ- 
ence of  i  upon  the  C.  E.  M.  F.  of  the  circuit. 

214.  Inductance  and  Capacitance.  —  Extending  this 
method  to  the  case  of  a  circuit  in  which  a  condenser  of 
capacity  C  farads  is  connected  with  the  sinusoidal  E.  M.  F. 
e  through  an  inductant  resistance  of  R  ohms  and  L  henrys, 

we  find  the  C.  E.  M.  F.  of  self-induction  to  be  —  L  -=-  volts, 

at 

and  the  C.  E.  M.  F.  of  capacity  to  be  —  -^J^  idt  volts,  while 

the  apparent  C.  E.  M.  F.  of  resistance  is  —  iR  volts.     The 
total  C.  E.  M.  F.  in  the  circuit  is  consequently 


-     iR  -f  Z        +  «»      VoltS, 

so  that,  equating  the  impressed  and  reversed  C.  E.  M.  F., 
we  have 

e  =  E  sin  tot  —  iR  +  L~  +  ^Jo  idt  volts,       .   (6) 
The  solution  of  this  equation  is 

-} 


-  i)' 


328  IMPEDANCE. 

The  effective  current  strength  will,  therefore,  be 

—          E  i 

/=  — -=:-. : — ITT  amperes.     .     .     (8) 


Plotting    this    case    graphically   in    Fig.   98,    OI    is   the 


r,.  M.  F.  iR,  active  in  producing  the  current  i  amperes, 
through  the  resistance  R  ohms.  IA  is  the  E.  M.  F.  active 
in  overcoming  the  C.  E.  M.  F.  of  self-induction,  and  AE  is 
the  E.  M.  F.  exerted  in  overcoming  the  C.  E.  M.  F.  of  ca- 
pacity, so  that  EA  is  the  C.  E.  M.  F.  of  capacity,  while  OE  is 
the  impressed  E.  M.  F. 

It  is  evident  from  the  preceding  that  even  with  so  simple 
a  circuit  as  that  comprising  a  sinusoidal  E.  M.  F.,  resistance, 
inductance  and  capacity,  the  differential  equation  which  has 
to  be  solved  in  order  to  evaluate  the  current  strength  is 
already  formidable,  and  when  any  further  degree  of  com- 
plexity is  introduced,  such  as  branch  circuits,  the  difficulty 
of  effecting  the  solution  of  the  problem  becomes  very  great. 

215.  Inductance  and  Reactance. — 2d.  The  alternative 
method  of  dealing  with  sinusoidal  alternating-current  cir- 
cuits is  to  assume  that  no  E.  M.  Fs.  act  in  the  circuits  other 
than  the  impressed  E.  M.  Fs,  In  other  words,  the 
C.  E.  M.  Fs.  of  inductance  and  capacity  are  ignored,  but  their 


IMPEDANCE.  329 

effects  are  considered  as  resistances  added  to  the  circuit  and 
opposed  to  the  impressed  E.  M.  Fs. 

Thus,  an  inductance  of  L  henrys  is  considered  as  a  react- 
ance of  Loo  ohms,  and  any  capacity  of  C  farads  is  considered 

i 

as  a  reactance  of  —  ~~*-  ohms."" 
CGO 

A  reactance  is  reckoned  in  a  direction  making  an  angle  of 
90°  with  that  of  resistance,  considered  as  a  definite  straight 
line  or  axis. 

Thus,  if  a  resistance  of  R  ohms,  Fig.  99,  be  connected  in 

A 


Loa 


if" 

FIG   99. 

circuit  with  an  inductance  of  L  henrys,  the  reactance  of  the 
latter  will  be  Loo  ohms,  and  the  combined  resistance  and  re- 
actance will  be  OR  +  RA,  where  OR  is  the  resistance  and 
RA  the  reactance.  Consequently,  OA  will  be  the  impe- 
dance, or  geometrical  sum  of  OR  and  RA.  If  j  denotes 
V—  i,  or  the  quadrantal  versor  operator,  RA  =.jLa>,  and 
the  impedance  OA 

Z  —  R  -}-jLoo  ohms (9) 

The  effective  current  strength  in  the  circuit  will  be  in 
amperes 

impressed  effective  E.  M.  F.  in  volts 
impedance  in  ohms 


where  E  is  the  effective  E,  M.  F. 


33°  IMPEDANCE. 

From  the  point  of  view  of  impedance,  therefore,  we  con- 
sider a  sinusoidal  current  circuit  as  acted  upon  by  impressed 
E.  M.  Fs.  only,  and  consider  the  resistance  as  modified  by 
the  presence  of  inductance  or  capacity. 

Taking  again  a  circuit  of  inductance,  resistance  and  ca- 
pacity, 

The  resistance  is  R  ohms ; 

The  reactance  of  inductance  isjLca  ohms ; 

The  reactance  of  capacity  is  —  -^—  ohms ; 

The  impedance  is  R  -\-J\LGO  —    ~ )  ohms  ; 

»  L,Gd ' 

so  that  the  current  strength  is 

E 
i  —  7 ~\  amperes (n) 


This  expression  is  equivalent  to  that  of  (8). 

When  a  sinusoidal  current  circuit  contains  any  number  of 
inductances,  resistances  and  condensers  in  series,  the  impe- 
dance is  the  geometric  sum  of  all  the  resistances  and  react- 
ances. That  is, 

1=  -  —. — amperes.     .     .     (12) 

^A.  — r-   7  \  ^J-^Gd    —    -^~7^ ] 

J  \  CooJ 

The  computation  may  be  carried  out  by  algebraic  rules  or 
by  geometry. 

It  is  to  be  observed  that  the  above  equation  is  a  versor 
equation,  and  that*/  is  no  longer  a  simple  number  of  amperes, 
but  a  number  of  amperes  turned  through  a  certain  angle. 
The  execution  of  the  operations  demanded  by  equation  (12), 
namely,  the  division  of  the  quantity  E  by  the  complex 
quantity  representing  the  impedance,  results  in  a  quotient  /, 


IMPEDANCE.  331 

which  is  also  a  complex  quantity  and  should  be  represented 
by  a  line  having  both  direction  and  magnitude,  the  direction 
being  confined  to  a  single  plane  and  to  an  angle  less  than 
90°  from  E.  .  $ 

216.  Joint  Impedances. —  When  sinusoidal  current  cir- 
cuits are  connected  in  parallel,  their  joint  impedance  pre- 
sents, symbolically,  no  more  difficulty  than  the  treatment  of 
joint  resistances  in  continuous-current  circuits.  Thus  if  Z^ , 
Z^ ,  Z3  .  .  .  Zn  be  the  impedances  of  n  circuits  connected  in 
parallel,  each  impedance  being  a  plane  vector,  or  versor 
quantity,  expressible  in  the  form 


Zn  =  Rn  +j\Ln(v  -  ^-^1  ohms,  ...     (13) 

then  the  reciprocal  of  Zn  may  be  called  the  admittance  of 
the  circuit  n.  The  admittance  of  a  circuit  is  a  plane  vector 
or  directed  quantity  expressible  in  mhos.  Thus  if  YH  be  the 
admittances  of  circuit  n, 

Yn  =  -^-=-         -7-          -—^  mhos.  .     .     (14) 

£*m  »- 


The  joint  admittance  of  the  group  of  multiple  circuits 
will  then  be  the  geometrical  sum  of  all  the  admittances,  or 


F= 


The  joint  impedance  will  be  the  reciprocal  of  the  joint 
admittance,  or 


Z  will  obviously  be  a  plane  vector  quantity. 


332  IMPEDANCE. 

When  any  inductant  resistance  in  a  sinusoidal  circuit  is 
linked  magnetically  with  a  secondary  circuit,  through  a  mu- 
tual inductance  of  L^  henrys,  the  presence  of  the  secondary 
circuit  will  modify  the  impedance  of  the  inductant  resistance 
in  a  definite  manner. 

Let  Rs  and  Xs  be  the  resistance  and  total  reactance  of  the 

secondary  circuit  ; 

RP  and  Xp  be  the  corresponding  resistance  and  react- 
ance of  the  inductant  resistance  forming  the 

primary;  then  if  n  =  —  ^-  ,  where 
•»« 

Zs  =  Rs  +JX,  , 

the  effective  impedance  of  the  inductant  resistance  acting  as 
the  primary  is  changed  from  Rp  -\-jXp  in  the  absence  of 
the  secondary  circuit  to 


Z=Rp  +  n*Rs  +  j(Xp  -  n*Xs)  ohms 
in  the  presence  of  the  secondary  circuit. 


?  •'  '• 

THE   PROPAGATION   OF   CURRENTS. 

GENERAL   CONSIDERATIONS. 

217.  Phenomena  which  Accompany  the  Propagation 
of  the  Current  in  a  Conductor. — Ohm  has  furnished  an 
expression  for  a  continuous  current  in  a  conductor,  relying 
on  an  assimilation  between  the  electric  and  the  calorific 
fluxes. 

If  we  call  r  the  resistance  of  a  conductor  per  unit  length, 

—  -TJ-  the  variation  of  potential  per  unit  length  in  the  di- 
rection /  of  the  conductor,  we  have 

dU 


Lord  Kelvin  has  carried  this  law  further  so  as  to  make  it 
applicable  to  the  variable  period  of  a  current  which  is  being 
set  up  in  a  conductor  of  given  capacity.  Imagine  an  in- 
sulated cable  submerged  in  water.  This  cable  forms  a 
cylindrical  condenser,  with  the  water,  supposed  to  be  at 
zero  potential,  as  its  outer  plate.  The  moment  a  difference 
of  potential  is  established  between  the  two  ends  of  the  con- 
ductor, a  current  is  set  up;  but  at  the  same  time  every 
portion  of  the  conductor  is  charged  with  a  quantity  of 
electricity  in  proportion  to  its  capacity  and  the  difference 
between  the  potentials  of  the  plates.  There  then  occurs, 
across  the  dielectric  of  the  cable,  a  charge,  or  displacement- 
current  (Maxwell),  which  may  be  considered  as  a  derived 
current  from  that  in  the  conductor.  This  displacement- 

333 


334  THE   PROPAGATION   OF  CURRENTS. 

current  ceases  when  the  tension  of  the  dielectric  balances 
the  potential-difference  of  the  condenser-plates. 

Then  there  exists  in  the  dielectric  an  electric  field  whose 
lines  of  force  connect  the  condenser-plates  and  end  in  equal 
and  opposite  quantities  of  electricity  (§  88). 

When  the  period  of  charge  is  past,  there  only  remains  the 
permanent  current  expressed  by  equation  (i). 

The  same  phenomenon  occurs,  but  in  a  lesser  degree, 
when  a  current  is  sent  through  a  circuit  in  the  air,  for  the 
charge  in  the  conductor  excites  a  contrary  charge  upon  the 
neighboring  conductors  separated  by  air  or  other  dielectric. 

Denote  by  c  the  capacity  of  the  conductor  per  unit  of 
length  ;  the  corresponding  charge  for  a  difference  of  poten- 
tial U  is 

q=cU.  ........     (2) 

The  energy  of  the  electric  field  per  unit  length  is  %cU*. 

Let  'us  express  the  fact  that  the  quantity  of  electricity 
which  enters  the  given  segment  of  the  cable  is  equal  to  that 
which  flows  out,  plus  the  current  across  the  dielectric.  The 
variation  of  the  current  in  the  conductor  per  cm  is  repre- 

sented by  —  -r    ;  the  displacement-current  is   —  ,  whence 


the  condition 


_  diL  =  dq^          dU 

dl       dt    ~Cdt (3) 


Combining  (i)  and  (3),  we  get  the  equation 


This  elementary  law  has  enabled  Lord  Kelvin  to  investi- 
gate the  variable  period  in  submarine  cables,  where  conden- 
ser-phenomena play  an  all-important  part. 


GENERAL    CONSIDERATIONS.  335 

It  must  be  noted  that  the  current  traversing  the  conduc- 
tor is  manifested  by  a  loss  of  energy  transformed  into  heat 
by  the  Joule  effect. 

The  displacement-current  ^n-the  dielectric  represents,  on 
the  contrary,  a  storage  of  energy  in  the  potential  state, 
shown  by  the  tension  of  the  dielectric.  This  energy  in  its 
turn  produces  heating  effects  when  the  cable  is  discharged. 

In  order  to  investigate  the  manifestations,  observed  dur- 
ing the  variable  period,  in  a  general  way,  it  is  necessary  to 
call  in  the  magnetic  phenomena  produced  by  the  current  in 
the  surrounding  medium. 

We  have  just  seen  that  the  current  develops  an  electric 
field  whose  intensity  depends  on  the  specific  inductive  ca- 
pacity of  the  dielectric  and  whose  lines  of  force  are  normal 
to  the  conductor.  But  besides  this,  the  current  creates  a 
magnetic  field  characterized  by  lines  of  force  forming  closed 
curves  around  the  conductor ;  the  intensity  of  this  field  is 
proportional  to  the  permeability  of  the  surrounding  medium 
magnetized  by  the  current. 

The  intensity  of  the  magnetic  field  diminishes  rapidly 
with  the  distance  from  the  conductor.  We  may  therefore 
say  that  the  self-induction  of  a  circuit  is  sensibly  propor- 
tional to  the  length  of  the  conductors  composing  it,  on  con- 
dition that  they  are  far  enough  apart  so  that  the  lines  of 
force  developed  by  them  do  not  encroach  on  each  other. 

Such  an  encroachment  occurs  in  the  case  of  two  wires 
stranded  together  and  connected  in  series.  They  tend, 
under  the  influence  of  an  electric  current,  to  produce  lines 
of  force  opposite  in  direction  in  the  surrounding  medium, 
so  that  their  coefficient  of  self-induction  is  practically  zero. 
The  same  holds  for  a  bobbin  wound  with  a  wire  bent  double 
(§  1 88). 

Putting  these  cases  aside,  denote  by  Ls  the  coefficient  of 
self-induction  of  a  circuit  per  unit  length  of  the  conductors. 


336  THE  PROPAGATION  OF  CURRENTS. 

The   magnetic   energy  of   the   current,  represented   by  the 
magnetization  of  the  medium,  is  \Lf  per  centimetre. 

The  medium  opposes  a  certain  inertia  to  magnetization, 
which  has  the  effect  of  developing  an  E.  M.  F.  contrary 
to  that  which  gives  rise  to  the  current.  This  E.  M.  F.  is 

—  Ls  —  per  centimetre,  so  that  Ohm's  formula  when  corri- 
da 

pleted  becomes 


Equations  (3)  and  (5)  enable  us  to  treat  the  problem  of 
the  variable  period  in  its  whole  extent,  taking  into  account 
the  production  of  the  electric  and  the  magnetic  fields  cre- 
ated by  the  current. 

We  have  supposed  the  current  to  be  surrounded  by  a 
perfect  insulator.  If  there  were  any  losses  of  electricity 
across  the  dielectric,  we  would  have  to  add  to  the  second 
member  of  the  preceding  equation  a  term  accounting  for 
the  derived  currents  due  to  the  electric  conductivity  of  the 
medium. 

Equations  (3)  and  (5)  are  analogous  to  those  met  with  in 
the  theory  of  the  propagation  of  sound-waves,  when  we 
admit  that  the  passive  resistance  of  the  medium  is  propor- 
tional to  the  first  power  of  the  velocity,  and,  moreover,  that 
electric  resistance  corresponds  to  friction,  self-induction  to 
the  inertia  of  the  medium,  and  capacity  to  the  reciprocal  of 
a  pressure. 

From  this  analogy  it  follows  that,  if  a  circuit  be  subjected 
to  a  periodic  E.M.F.,  the  electric  waves  generated  are  propa- 
gated in  accordance  with  laws  identical  with  those  of  the 
propagation  of  sound.  In  particular,  if  a  periodic  E.  M.  F.  is 
applied  to  one  of  the  ends  of  a  line  insulated  at  the  other 
end,  the  electric  waves  thus  created  are  reflected  at  the  insu- 
lated end  and  return  to  their  starting-point,  where  they  are 


GENERAL    CONSIDERATIONS.  337 

reflected  again,  just  like  sound-waves  sent  into  a  tube  which 
is  closed  at  one  end. 

The  analogy  noticed  above  is  of  particular  interest  in 
telephony,  for  it  shows  that  "the  electric  waves  transmit 
speech  according  to  laws  identical  with  those  which  govern 
its  propagation  in  a  ponderable  medium.* 

218.  Special  Characteristics  shown  by  Alternating 
Currents. — As  has  been  shown  in  §  199,  alternating  cur- 
rents do  not  add  together  as  do  continuous  currents,  but  are 
compounded  like  vectors  according  to  the  parallelogram  of 
forces  :  it  is  for  this  reason  that  two  equal  periodic  E.  M.  Fs. 
acting  in  a  circuit  do  not  usually  give  a  resultant  current 
double  that  which  could  be  produced  by  each  one  of  them. 
The  mean  resultant  is  not  equal  to  the  sum  of  the  mean 
component  currents  unless  the  phases  of  the  latter  are  to- 
gether; it  is  zero  if  the  phases  differ  by  180°,  just  as  the 
resultant  of  two  forces  is  zero  when  they  are  equal  and  in 
opposite  directions. 

This  view  explains  certain  peculiar  effects  produced  by 
periodic  electromotive  forces. 

Consider,  for  example,  an  alternating  current  split  into 
two  branches  having  different  resistances  and  self-induc- 
tions.f  The  branch  currents  will  present  differences  of 
phase  with  regard  to  each  other  and  to  the  total  current, 
§  198;  at  each  instant  the  total  current  will  be  equal  to 
the  sum  of  the  branch-currents,  but  the  mean  total  current 
will  by  no  means  be  equal  to  the  sum  of  the  mean  branch- 
currents.  If  the  difference  of  phase  of  the  latter  is  great 
enough,  it  may  even  happen  that  the  mean  current  of  each 
one  of  the  branches  will  be  greater  than  the  whole  current. 

*  See  Demany,  Thdorie  de  la  propagation  de  Velectricitt.  Bulletin  de 
L'association  des  ingenieurs  sortis  de  1'Institut  Montefiore,  1890. 

f  Lord  Rayleigh,  On  Forced  Harmonic  Oscillations.    Phil.  Mag.,  May,  1886. 


THE   PROPAGATION   OF  CURRENTS. 

It  is  sufficient,  for  this  purpose,  to  have  the  angle  of  lag 
more  than  120°,  for  the  parallelogram  of  forces  shows  that 
when  two  equal  vectors  make  an  angle  of  120°,  their  result- 
ant is  equal  to  one  of  the  component  vectors.  The  sine- 
curve  representing  the  total  current  will  have,  at  each 
instant,  its  ordinates  equal  to  the  algebraic  sum  of  the  ordi- 
nates  of  the  two  component  sine-curves.  These  latter  may 
therefore  have  very  much  greater  coordinates  of  the  result- 
ant curves. 

Take  another  case,  pointed  out  by  M.  Smith.  Between 
two  points  a  and  d  subjected  to  an  alternating  potential- 
difference  of  constant  mean  value,  put  two  resistances  ab 
and  bd  in  series,  one  having  a  considerable  coefficient  of 
self-induction,  the  other  non-inductive.  On  each  resistance 
put  a  glow-lamp  on  a  shunt.  The  two  lamps  are  alike,  and 
the  non-inductive  resistance  is  adjusted  until  they  both  burn 
with  the  same  brilliancy;  this  result  shows  that  equal 
currents  are  traversing  the  filaments,  and  that  the  potential- 
difference  of  the  points  #,  b  is  the  same  as  that  of  b,  d.  Next, 
the  wire  joining  the  lamps  is  separated  from  that  connecting 
the  resistances,  and  the  brilliancy  of  the  lamps  is  seen  to  de- 
crease, although  the  potential-difference  of  the  points  a,  d 
retains  the  same  mean  value  ;  which  proves  that  this  latter  is 
less  than  the  sum  of  the  mean  differences  found  between  the 
points  a,  b  and  b,  d.  This  fact  is  also  explained  by  a  differ- 
ence of  phase  in  the  potential-differences  to  which  the  two 
resistances  in  series  are  subjected  :  as  the  resultant  has  re- 
mained constant,  the  two  components,  whose  phases  are  not 
together,  must  have  assumed  values  greater  than  half  the 
acting  potential-difference. 

Instead  of  using  glow-lamps  to  indicate  potential-differ- 
ences, we  might  have  used  the  quadrant  electrometer  as 
arranged  in  §  102  (II). 

It  is  unnecessary  to  state  that  there  is  no  contradiction  of 


GENERAL    CONSIDERA7UONS.  339 

the  principle  of  the  conservation  of  energy  in  these  experi- 
ments. If  we  find  an~  increase  of  power  consumed  in  one 
branch  of  a  circuit,  we  observe  a  corresponding  expenditure 
at  the  source  of  electricity,  by  the  increase  in  the  mean 
power  of  this  source, 

Pm  =  Eeff  .Ieff.  cos0. 

219.  Comparative  Effects  of  the  Self-induction  and 
Capacity  of  a  Circuit. — It  is  interesting  to  observe  that  in 
considering  the  flux  of  electricity  which  flows  during  the 
variable  period,  the  capacity  and  self-induction  play  opposite 
parts.  In  fact,  the  displacement-current  due  to  the  capacity 
is  added  to  the  current  which  traverses  the  conductor,  so 
that  the  phenomenon  of  condensation  is  equivalent  to  an 
apparent  diminution  of  the  resistance  of  the  circuit  during 
the  variable  period. 

Suppose  a  conductor  of  resistance  r,  whose  ends  are  at  a 
potential,  one  of  U,  the  other  zero.  Insert,  between  the 
extremities  of  the  conductor,  a  condenser  of  capacity  c\  its 
charge  at  the  end  of  the  variable  period  is  q  =  cU  =  dr. 
This  charge  is  added  to  the  quantity  of  electricity  traversing 
the  circuit. 

On  the  contrary,  the  electromagnetic  induction  produces 
an  increase  of  the  impedance  and  a  diminution  of  the  flux 
of  electricity,  during  the  variable  period  of  closure,  equal  to 

(§  194) 


From  these  opposite  effects  follows  a  certain  compensation 
which  can  be  made  use  of  in  the  transmission  of  signals  in 
cables. 

The  difference  of  the  fluxes  of  the  extra-current  and  dis- 
placement-current is 

Lsi  ilr  \ 

q  —  q  —  —  —  cir  —  -{ L8  —  cr  1. 
r  r\  I 


340  THE  PRO  PAG  A  TION  OF  CURRENTS. 

We  see  that,  as  regards  the  flux  of  electricity  transmitted 
during  the  variable  period,  the  effect  of  the  condensation 
corresponds  to  a  diminution  of  the  self-induction  equal  to 
the  product  of  the  capacity  by  the  square  of  the  resistance 
of  the  conductor.  This  observation  is  due  to  Sumpner. 

220.  Effect  of  a  Capacity  in  a  Circuit  Traversed  by 
Alternating  Currents.*— A  condenser  may  be  inserted 
in  series  in  a  circuit  traversed  by  alternating  currents  with- 
out interrupting  the  passage  of  these  currents,  as  would  be 
the  case  if  we  were  dealing  with  a  continuous  E.  M.  F.  In 
fact,  at  each  reversal  of  the  current  the  condenser  is  dis- 
charged, and  recharged  in  the  opposite  direction.  It  is  nec- 
essary, however,  if  a  powerful  mean  current  is  desired,  that 
the  capacity  of  the  condenser  should  be  great  enough  to 
absorb  the  electric  flux  transported  by  the  waves  of  the 
current. 

Suppose  that  in  a  circuit  of  resistance  r  and  without 
self-induction,  we  insert  a  condenser  of  capacity  c. 

Denoting  by  E  —  EQ  sin  at  the  periodic  E.  M.  F.,  and  by 
u  the  potential-difference  of  the  plates  at  an  instant  /,  we 
have 

E  =  E0  sin  at  —  u  -f-  ri.  .    .     .     .     .     (i) 

But 

cdu  =  tdt.     .    .    .-.-    .    V    .     .     (2) 

Differentiating  (i)  and  substituting  for  du  its  value  ob- 
tained from  (2),  we  arrive  at 

a£0  cos  at  d/  =  -  d/  +  rdt.     ....     (3) 

This  equation  has  a  form  similar  to  equation  (3)  of  §  1 80. 
Resolving  it  by  the  same  method  and  suppressing  the  ex- 
ponential term,  which  becomes  zero  at  the  end  of  a  very 

*  Boucherot,  Electricien,  Nov.  15,  1890. 


GENERAL    CONSIDERATIONS.  34! 

short    time,  we    find    for    the    expression    of    the    regular 

current 

£ 
i  —  —  — .sin  (at  +  0)     ....     (4) 

i/r'4__L_" 

-2gi 

on  condition  that 

0  =  tan"1 .     ......     (5) 

acr 

This   result   shows   that   the    capacity  has  the  effect  of 
advancing  the  phase  of  the  current  ahead  of  the  E.  M.  F. 
We  have  moreover 


«V 


which  proves  that  the  condenser  reduces  the  current  the 
more,  the  less  its  capacity  is.  The  current  becomes  zero 
for  c  =  o.  An  infinite  capacity  would  produce  the  same 
effect  as  taking  the  condenser  away. 

We  could  have  found  the  preceding  equations  directly  by 

substituting,  in  the  equations  of  §. 1 80,  — -  for  the  factor 
aLs.  This  fact  is  explained  if  it  is  observed  that  the  self- 
induction  introduces  an  electromotive  force  e  =  —  AT- , 

while  a  condenser  brings  in  a  difference  of  potential  (taken 
from  (2) ) 

e'  =  -  u  =  -  -  -  A'd/. 

Substituting  for  i  its  value  /  sin  at  in  these  expressions, 

we  find 

e   =  —  aLJ  cos  at, 

e'  =  -\-  — /  cos  at ; 
ac 

values  which  are  equal  if  we  put  aLs  = . 


342  THE  PROPAGATION  OF  CURRENTS. 

221.  Combined  Effects  of  a  Capacity  and  a  Self- 
induction  in  a  Circuit  Traversed  by  Alternating  Cur- 
rents. Ferranti  Effect.  —  If  we  introduce  in  a  circuit 
traversed  by  alternating  currents  a  self-induction  and  a 
capacity,  as  the  first  tends  to  make  the  phase  of  the  current 
lag,  and  the  second  tends  to  advance  it,  there  will  be  a  more 
or  less  complete  neutralization  of  the  two  effects. 

Let  us  analyze  this  combination  :  Keeping  the  preceding 
notation,  we  will  have  the  two  equations 

E  =  EQ  sin  at  =  A^  +  ri+u,      .     .     .     (i) 

cdu  =  tdt,     .......     (2) 

and,  combining  them, 

Lsc-^+rc~  +  i-  E,accvsat  =  o.      .     .     (3) 

The  general  solution  of  this  differential  equation  is  of  the 
form 

i  =  Aemt  +  B  sin  at  +  D  cos  at.     .     .     .     (4) 

Differentiating  this  equation   twice    and    introducing  in 

di          dV 
(3)  the  values  of  /—  and  -7-3  thus  found,  we  can  determine 

the  arbitrary  coefficients  m,  B,  and  D.  If  we  notice  more- 
over that  the  exponential  term  is  very  quickly  zero  with  /, 
the  value  of  the  established  current  /  reduces  to 

*  =  -  ---  -  '  -  --     sin  (at  -  0)       o     .     (5) 


on  condition  that 


,A--     :.. 

0=tan-'  -  ~.     ....     (6) 


GENERAL    CONSIDERATIONS.  343 

The  effective  current  is 

% 

's  ~~^~r) 


ac 

m 


This  expression  could  have  been  found  directly  by  sup- 
posing a  circuit  with  two  self-inductions  Ls  and  Z/  and 
substituting  for  the  second  one  a  capacity  such  that 


Equation  (6)  shows  that  the  current-phase  will  be  behind 
or  ahead  of  the  E.  M.  F.  phase  according  as  aL,  is  larger 

or  smaller  than  —  ,  or  a*Lsc^  I.     In  every  case  the  current 

will  be  less  than  if  there  were  neither  self-induction  nor 
capacity,  unless  d*Lc  —  I. 

If  we  determine  from  equation  (5)  the  expression  for  the 
E.  M.  Fs. 


e  =  —  Ls  -r-     and     ef  — /  idt 

at  cj 

due  to  the  self-induction  and  the  capacity,  we  find 

€    =    -A 


It  will  be  noted  that   these   values    may   be  very  much 

higher  than    E ;    in    particular,    if   aLs  =  —    and    r   tends 

ac 

towards  o,  it  is  easily  seen  that  the  potential  differences  at 
the  terminals  of  the  induction-coils  and  the  condenser  tend 


344  THE  PROPAGATION   OF  CURRENTS. 

towards  infinity.  These  peculiarities  are  explained  by  the 
composition  of  the  E.  M.  Fs.,  as  has  been  seen  in  §  218. 

It  is  to  a  similar  cause  that  the  effect  must  be  attributed 
which  was  observed  in  Ferranti's  cables  in  London  :  when 
these  cables,  which  are  concentric  and  have  a  considerable 
capacity,  are  traversed  by  alternating  currents,  an  elevation 
of  tension  is  observed  at  the  end  of  the  line. 

From  all  the  above  deductions  it  follows  that  the  capacity 
of  a  circuit  corrects  the  effects  of  the  self-induction  by  de- 
creasing the  impedance  created  by  the  latter,  as  well  as  the 
phase-lag.  The  capacity  to  give  a  circuit  in  order  to  com- 
pletely neutralize  the  effects  of  its  self-induction  is  shown 
by  the  equation  c?Lsc  —  I.  If  a  =  100,  for  example,  and 
Ls  =  one  quadrant,  we  will  have  c  =  100  microfarads.  This 
capacity,  which  can  only  be  obtained  by  the  use  of  expensive 
condensers,  could  be  reduced  by  artificially  increasing  the  self- 
induction  by  electromagnets  with  closed  magnetic  circuits. 

The  problem  of  the  subdivision  of  a  periodic  current  in  a 
branch  having  self-induction,  and  another  with  a  condenser, 
also  gives  rise  to  interesting  observations.  The  current  in 
the  inductive  resistance  will  have  a  phase-lag  behind  the 
resultant  current,  while  the  flux  in  the  condenser  will  be  ad- 
vanced. Consequently  the  sum  of  the  two  derived  fluxes 
will  be  greater  than  the  resultant  flux,  and  each  of  the 
branches  may  even  be  traversed  by  a  mean  current  superior 
to  the  total  current. 

222.  Oscillating  Discharge. — Let  us  take  up  again  the 
discharge  of  a  condenser,  by  investigating  which  we  ap- 
proached the  study  of  the  electric  current  (§  103),  being 
guided  by  our  knowledge  of  the  laws  of  induction. 

Suppose  a  condenser  of  capacity  c,  whose  plates  are  at  a 
difference  of  potential  U.  Call  r  and  Ls  the  resistance  and 
self-induction  of  the  circuit  of  discharge.  It  is  well  to  note 


GENERAL    CONSIDERATIONS.  345 

that  r  expresses  the  metallic  resistance  of  the  discharge- 
circuit. 

The  discharge-current  is  equal  to  the  rate  of  variation  of 
the  charge,  or  f  « 


but  q  =.  cU,  whence 


and 


To  resolve  this  equation,  put  q  =  emf,  and  we  get 


The  general  integral  is  of  the  form 

q  •=  Aem*'  +  Be**,  ......     (l) 

A  and  B  being  constants  of  integration;  ml  and  mt,  the 
roots  of  the  equation  obtained  by  equating  to  zero  the  tri- 
nomial in  parenthesis,  or 


2L, 

Substituting  these  values  in  (i)  and  putting  —  =  r,  we  get 

V  '  -  —  . 


__. 


If  the  roots  of  the  trinomial  are  imaginary,  the  radical 
exponent  takes  the  form 


_ 

err       4r8 


346  THE  PROPAGATION  OF  CURRENTS. 

and  equation  (2)  reduces,  by  Ruler's  formula,  to 


The  constants  of  integration  are  determined  by  the  simul- 
taneous conditions 

/  =  o,     i  =  o,     q  =  Q 
introduced  in  equations  (2)  or  (3)  and  in  the  expression 


Then  substituting  in  this  last  the  values  found,  we  get,  in 
the  case  of  real  roots, 


i  =  —  e    2TV  -  e  j.  (4) 


4T*       err 

In  the  case  of  imaginary  roots,  the  value  of  the  current 
assumes  the  form 

i  = JL=r^  sin /-L --!.,.      .    (5) 

T       4r3        •   - 


Equation  (4)  shows  that,  for 


or 


the  discharge  occurs  in  the  form  of  a  continuous  current, 
constant  in  direction,  beginning  with  a  value  of  zero,  rising 
rapidly  to  a  maximum,  and  then  rapidly  decreasing. 
When 


the  discharge-current  oscillates  periodically  between  positive 
and  negative  values  which  decrease  rapidly. 


GENERAL    CONSIDERATIONS. 


347 


Equation  (5)  shows  that   the  oscillating  current   passes 
through  the  same  phases  for 


—  ..t  =j  d,     27r, 


err 


Hence  we  deduce  that  the  period  of  the  oscillating  cur- 
rent is 

27T  27T 


T_ 


If  r  is  negligible  compared  with  2\  —  ,  we  get 


If,  for  example,  k=  i  microfarad,  Lf  =  i.io"4  quadrant, 


T  =  2n  Vi.io~°  X  i.io~*  =  0.000063  second. 

Figures  100,  101,  and  102  give  a  graphical  representation 
of  the  above  phenomena. 

The  curve  (I)  represents  the  variations  of  the  charge  in 
terms  of  the  time  in  the  case  of  a  continuous  discharge,  and 


FIG.  100. 


curve  (II)  shows  the  variations  of  the  current  which,  zero  at 
the  beginning  of  the  discharge,  rapidly  assumes  a  maximum 
value  and  then  decreases  asymptotically  towards  zero. 


348 


THE  PROPAGATION  OF  CURRENTS. 


Fig.  101  shows  the  variations  of  the  charge  in  the  case  of 
an  oscillating  discharge,  and   Fig.    102  shows  the  current- 


FIG.  IOT. 


variations.    These  variations  are  shown  by  decreasing  undu- 
lations whose  period  is  equal  to  T. 

We  are  indebted  to  Lord  Kelvin  for  the  first  analytical 


TIME        X 


FIG.  102. 

investigation  of  the  oscillatory  discharge,  which  of  late 
years  has  been  the  subject  of  experiments  performed  by, 
among  others,  Hertz  and  Lodge. 

To  understand  these  phenomena,  we  must  recall  the  fact 
that  the  dielectric  of  a  charged  condenser  is  subjected  to  a 
tension  which  may  be  compared  to  that  of  a  spring.  If  the 
cause  which  produces  the  tension  disappears  suddenly,  the 


GENERAL    CONSIDERATIONS.  349 

dielectric  returns  to  its  original  position  after  having  per- 
formed oscillations  comparable  to  those  of  a  spring  when 
suddenly  released. 

To  keep  the  spring  from  oscillating  we  must  oppose  a  re- 
sistance to  its  movement,  by  plunging  it,  for  example,  in  a 
viscous  fluid.  Thus,  too,  by  presenting  an  electrical  resist- 
ance to  the  discharge  of  a  condenser,  it  is  rendered  contin- 
uous. 

The  period  of  the  oscillations  of  a  spring  depends  upon 
its  mass  or  its  inertia.  So  the  period  of  the  electric  dis- 
charge varies  with  the  self-induction,  which  represents  the 
magnetic  inertia  of  the  medium  surrounding  the  circuit. 

By  increasing  the  self-induction  we  increase  the  length  of 
the  period  more  and  more  ;  to  reach  this  result,  it  is  suffi- 
cient to  pass  the  discharge  through  a  bobbin,  the  number  of 
whose  turns  increases  progressively.  Curiously  enough,  it 
is  useless  to  put  an  iron  core  in  the  bobbin,  for,  in  conse- 
quence of  the  rapidity  of  the  reversals  of  the  discharge,  the 
result  is  zero  as  far  as  affects  the  magnetization. 

By  proceeding  in  this  way,  i.e.,  by  interposing  an  increas- 
ing number  of  turns  of  wire  between  the  condenser-plates, 
we  get  an  oscillating  discharge  of  increasing  period.  The 
discharge-spark,  which  occurs  at  every  break  in  the  conduc- 
tors, appears  single  on  account  of  the  rapidity  of  the  phe- 
nomenon, but,  if  it  is  reflected  from  a  rotating  mirror,  we 
see  that  it  seems  composed  of  a  succession  of  luminous 
points.  On  increasing  the  period  sufficiently,  the  chain  of 
luminous  points  becomes  visible  to  an  observer  who  looks  at 
the  spark  through  a  glass  which  he  moves  rapidly.  Finally, 
when  the  self-induction  of  the  discharge-circuit  and  the 
capacity  of  the  condenser  are  sufficiently  great,  the  impulses 
communicated  to  the  air  by  the.  electric  undulations  reach 
the  limit  of  vibrations  perceptible  by  the  ear,  and  the  spirals 
give  a  note  whose  height  can  be  diminished  at  will. 


350  THE  PROPAGATION  OF  CURRENTS. 

Dr.  Lodge,  by  suitably  varying  the  values  of  r,  k,  Z5,  has 
succeeded  in  obtaining  a  scale  of  electrical  vibrations  whose 
periods  extended  from  one  hundred-millionth  to  one  five- 
hundredth  of  a  second. 

223.  Transmission  of  Electric  Waves  in  the  Surround- 
ing Medium. — The  comparison  of  the  oscillations  of  an 
electric  discharge  with  those  of  a  vibrating  elastic  body  can 
be  carried  still  further.  A  tuning-fork  generates  sound- 
waves in  the  surrounding  air,  which  spread  out  in  space  and 
can  be  shown  to  be  present  by  means  of  a  resonator  tuning- 
fork  in  pitch  with  the  first.  If  sound-waves  hit  perpendicu- 
larly against  a  solid  wall,  they  are  reflected  ;  the  incident 
waves  interfere  with  those  sent  back  from  the  wall,  and 
nodes  and  loops  of  vibration  are  formed  which  follow  each 
other  alternately.  The  resonator  remains  mute  at  the  nodes 
and  marks  the  position  of  the  loops  loudly.  The  length  of 
the  wave  is  equal  to  double  the  distance  between  two  con- 
secutive nodes.  The  velocity  of  transmission  of  the  waves 
is  the  quotient  of  the  length  of  a  wave  by  its  duration. 

Analogous  phenomena  appear  in  the  propagation  of 
calorific  and  luminous  radiations,  formed  by  vibrations 
transverse  to  the  direction  of  the  rays. 

Dr.  Hertz  has  experimentally  demonstrated  that  dis- 
charge-oscillations between  electrified  bodies  produce,  in 
the  surrounding  medium,  electric  waves  whose  properties 
are  identical  with  those  of  the  radiations  emitted  by  bodies 
at  high  temperatures,  that  is  to  say  that  they  give  rise  to 
phenomena  of  reflection,  interference,  refraction,  polariza- 
tion, and  diffraction.  In  order  to  exhibit  these  properties, 
it  was  necessary,  first  of  all,  to  have  an  apparatus  producing 
continual  electric  oscillations. 

Dr.  Hertz  made  use  of  a  Ruhmkorff  coil  for  this  purpose, 
Fig.  103,  the  secondary  terminals  of  which  are  connected 


GENERAL    CONSIDERATIONS 


351 


to  two  conductors  which  constitute  an  electric  vibrator  or 
exciter  and  which  have  been  given  a  number  of  shapes. 
In  Fig.  103,  the  vibrator  V  is  formed  by  two  conductive 
rods,  situated  in  the  line  of  each  other's  prolongation  and 
terminated  at  their  ^adjoining  ends  by  metallic  knobs,  the 
opposite  ends  carrying  metal  spheres.  These  latter  can 
be  replaced  by  discs  or  by  metal  plates  hung  from  the 
rods.  The  induction-coil  is  intended  to  keep  up  the  elec- 
tric charges  on  the  two  parts  of  the  vibrator.  In  virtue 


u    i 


FIG.  103. 

of  these  alternately  opposed  charges,  the  medium  sur- 
rounding the  vibrator  is  the  seat  of  an  electric  field  whose 
lines  of  force,  periodically  reversed  in  direction,  connect  the 
two  parts  of  the  apparatus.  The  mean  direction  of  the 
field,  that  is  to  say  of  the  electric  forces,  is  the  axis  of  the 
conductive  rods.  In  consequence  of  the  high  potential  of 
the  charges,  they  recombine  under  the  form  of  sparks  leap- 
ing between  the  two  metallic  knobs.  The  oscillations  of 
this  prolonged  discharge  are  dependent  on  the  capacity  of 
the  vibrator  and  the  resistance  and  self-induction  of  the 
conductors  composing  it:  note  that  here  the  resistance 
in  question  is  that  of  the  conductors  connected  to  the 
spheres.  When  the  charge  of  these  latter  is  sufficient,  a 
spark  occurs  between  the  adjoining  knobs,  and  it  is  only 
then  that  electric  oscillations  take  their  rise  between  each 
sphere  and  the  corresponding  knob.  The  duration  of  these 
oscillations  is  very  small  on  account  of  their  rapid  decrease, 


352  THE  PROPAGATION   OF  CURRENTS, 

Fig.  IOI  ;  the  induction-coil  must,  therefore,  renew  the 
charges  of  the  spheres  very  rapidly,  and  the  sparks  succeed 
each  other  at  very  short  intervals. 

The  oscillations,  which  produce  rapid  alternating  currents 
in  the  rods  of  the  vibrator,  are  propagated  in  the  surround- 
ing medium. 

To  demonstrate  this,  Dr.  Hertz  made  use  of  an  apparatus 
capable  of  vibrating  in  unison  with  the  electric  vibrator  and 
which  is  called  an  electric  resonator. 

The  resonator  can  be  identical  with  the  vibrator,  as  in 
Fig.  103,  or  else  it  may  have  an  entirely  different  form, 
provided  that  the  three  characteristic  magnitudes,  capacity, 
resistance,  and  self-induction  of  the  conductors  composing 
it,  satisfy  the  same  conditional  equation  as  the  correspond- 
ing quantities  of  the  vibrator, — a  fact  which  can  also  be 
experimentally  verified. 

This  being  so,  if  we  place  the  resonator  in  the  vicinage  of 
the  vibrator  so  that  their  axes  are  parallel,  a  flow  of  sparks 
is  observed  between  the  knobs  of  the  first-named  apparatus. 
These  sparks,  caused  by  the  variations  of  the  electric  force 
parallel  to  the  axis  of  the  conductors,  diminishes  steadily  as 
the  resonator  is  moved  further  off.  The  explosive  distance 
naturally  decreases  with  the  maxima  of  the  electric  force. 

The  phenomenon  of  electric  resonance  is  produced  even 
if  a  solid  insulating  wall  is  interposed  between  the  vibrator 
and  the  resonator,  e.g.,  a  vertical  partition  of  wood  or 
masonry;  but  it  ceases  when  the  partition  is  a  conductor. 
In  this  latter  case,  if  the  resonator  be  moved  between  the 
vibrator  and  the  partition,  it  is  found  that  in  certain  points 
the  spark  ceases  and  that  in  others  it  is  re-enforced. 

These  extinctions,  which  are  repeated  at  intervals  of  equal 
length,  prove  that  the  transmission  of  electric  forces  takes 
place  in  the  form  of  waves  capable  of  being  reflected  by  a 
conductive  wall,  the  waves  sent  back  by  the  wall  being 


GENERAL    CONSIDERATIONS.  353 

capable  of  interfering  with  the  incident  waves  to  form  the 
observed  nodes  and  loops  of  vibration.  The  distance 
between  two  nodes  corresponds  to  half  a  wave-length. 

To  obtain  a  number  of>  rfodes  in  the  limits  of  the  room 
used  for  experimenting  in,  i^is  necessary  to  produce  short 
waves.  By  using  as  vibrator  a  brass  tube  26  cm.  long  and 
3  cm.  in  diameter,  divided  into  two  parts  with  the  adjoining 
ends  terminating  in  segments  of  spheres,  Hertz  succeeded 
in  exciting  waves  only  30  cm.  in  length.  The  resonator  was 
a  straight  wire  I  metre  long,  also  divided  in  two  parts  and 
furnished  with  metallic  knobs  between  which  the  sparking 
takes  place. 

Hertz  has  likewise  performed  a  very  interesting  series  of 
experiments  with  the  aid  of  a  metallic  parabolic  reflector 
designed  to  concentrate  the  electric  undulations  in  a  given 
direction  and  to  thus  produce  more  marked  effects. 

The  reflector  is  composed  of  a  sheet  of  zinc  2  metres  in 
height,  curved  and  held  on  a  wooden  frame  so  as  to  have 
the  form  of  a  cylindrical  surface  having  a  parabolic  section. 
The  focal  line  is  then  parallel  to  the  generatrices  of  the 
cylinder. 

If  the  vibrator  is  placed  so  that  its  axis  coincides  with 
the  focal  line,  the  waves  are  propagated  in  the  direction  of 
the  reflector's  plane  of  symmetry  and  are  perceptible  by  the 
resonator  at  a  much  greater  distance  than  before.  The 
distance  at  which  they  can  be  perceived  is  still  further 
increased  by  placing  the  resonator  along  the  focal  line  of  a 
reflector  similar  to  the  first,  so  situated  that  the  planes  of 
symmetry  of  the  two  coincide. 

An  insulating  screen,  such  as  a  wall,  placed  between  the 
two  parabolic  mirrors,  does  not  intercept  the  undulations ; 
but  a  conducting  screen  stops  them  and  casts  a  shadow 
behind  it.  It  will  be  observed  that  we  are  naturally  led 


354  THE  PROPAGATION  OF  CURRENTS. 

to  employ  the  language  of  optics  to  denote  electric  un- 
dulations. 

These  phenomena  are  identical  with  those  produced  by 
luminous  rays ;  the  only  difference  is  in  the  order  of 
magnitude  of  the  wave-lengths;  luminous  waves  are  nearly 
a  million  times  shorter  than  the  electric  waves  produced  by 
means  of  the  vibrators  used  by  the  German  scientist.  The 
fact  that  Hertzian  vibrations  are  not  reflected  by  a  wall 
may  be  compared  to  the  fact  that  light  is  not  reflected  by 
a  very  thin  transparent  body :  we  know,  indeed,  that  soap- 
bubbles  no  longer  reflect  the  surrounding  objects  at  the 
moment  of  their  bursting ;  now  the  ratio  of  the  length  of 
light-waves  to  the  thickness  of  the  bubble  is  then  of  the 
same  order  as  the  ratio  of  the  lengths  of  the  Hertzian  waves 
to  the  thickness  of  a  wall. 

Electric  undulations  are  propagated  in  a  straight  line,  as 
is  shown  by  the  fact  of  their  stoppage  by  a  metal  screen. 
The  mode  of  producing  vibrations  shows  that  they  are 
parallel  to  the  axis  of  the  vibrator,  i.e.,  that  they  are  trans- 
versal and,  to  use  the  corresponding  optical  expression, 
polarized  rectilinearly. 

If  we  place  in  the  path  of  the  rays  transmitted  by  the 
first  mirror  a  screen  made  of  parallel  wires,  the  effect  of 
this  screen  is  very  different  according  as  the  wires  are 
parallel  or  perpendicular  to  the  axis  of  the  vibrator.  In  the 
first  case  the  electric  waves  pass  without  difficulty ;  in  the 
second,  the  electric  force  is  absorbed  by  the  wires  which  are 
normal  to  it  and  the  rays  are  extinguished.  The  effect  is 
similar  to  that  of  a  tourmaline  plate  in  optics. 

If,  after  having  removed  the  screen,  we  turn  the  resonator 
and  its  mirror  by  an  angle  of  90°  about  the  direction  of 
the  rays,  no  sparks  are  observed.  But  if  the  above-men- 
tioned screen  be  interposed  normally  to  the  transmitted 
rays  and  inclining  the  wires  45°  to  the  directions  of  the 


GENERAL    CONSIDERATIONS.  355 

focal  lines  of  the  mirrors,  the  screen  decomposes  the 
incident  waves  and  lets  the  vibrations  inclined  45°  to  the 
axis  of  the  resonator  go  past. 

These  can  then  act  on  the" resonator.  This  phenomenon 
recalls  the  lighting  up  of  tlsue  field  of  two  crossed  Nicol's 
prisms  by  the  interposition  of  a  crystal  plate. 

Lastly,  the  two  mirrors  enable  us  to  exhibit  cleaijy  the 
phenomenon  of  the  reflection  and  refraction  of  electric 
waves.  For  example,  if  we  send  an  electric  ray  against  a 
plane  conductive  partition,  we  can  perceive  the  reflected 
waves  by  the  aid  of  a  resonator,  provided  that  the  planes  of 
symmetry  of  the  two  parabolic  mirrors  be  placed  so  that 
they  intersect  each  other  at  the  partition  and  that  the 
normal  plane  through  their  intersection  makes  two  equal 
plane  angles. 

To  produce  refraction  Hertz  made  use  of  a  large  asphalt 
prism  having  a  refringent  angle  of  30°.  The  incident  rays 
directed  upon  the  prism  by  the  vibrator  make  with  the 
refracted  rays,  which  are  received  in  the  resonator,  an 
angle  indicating  an  index  of  refraction  1.7,  only  a  slightly 
higher  value  than  that  given  by  optical  experiments.  Hertz- 
ian waves  are  therefore  refracted  by  an  insulating  prism  as 
light  waves  are  by  a  glass  prism. 

These  experiments  by  Hertz  have  been  repeated  by 
various  physicists.  MM.  Sarrazin  and  de  la  Rive,  of  Ge- 
neva,* have  shown  that  the  form  of  the  resonators  can  be 
modified  without  ceasing  to  perceive  the  sparks,  and  that 
the  observed  wave-length  depends  much  more  on  the  dimen- 
sions of  this  apparatus  than  on  those  of  the  exciter, — which 
seems  to  show  that  the  latter  gives  rise  to  complex  waves 
which  can  be  picked  out  by  suitable  resonators.  This  ex- 
plains why,  when  an  oscillating  discharge  is  kept  up  in  a 

*  Archives  de  Geneve,  June,  1890. 


THE  PROPAGATION  OF  CURRENTS. 

room,  sparks  are  seen  to  fly  between  metallic  objects  which 
are  near  together. 

Lecher  of  Vienna  has  investigated  the  propagation  of 
Hertzian  waves  along  two  parallel  conductors,  placing  over 
their  ends  a  Geissler  tube  to  serve  as  a  resonator.  When  a 
metallic  bridge  is  slid  along  the  conductors,  the  brilliancy  of 
the  tube  grows  less  for  certain  positions  of  the  bridge,  and 
stronger  for  others.  This  phenomenon  is  comparable  to  the 
propagation  of  sound  in  a  closed  tube  ;  at  the  nodes  the 
pressure  of  the  air  is  different  from  the  atmospheric  pressure, 
at  the  loops  it  is  equal  to  this  pressure.  Consequently  the 
sound  is  not  altered  if  the  tube  be  pierced  at  a  loop,  but  it 
changes  if  the  orifice  is  opposite  a  node.  Thus  at  the  nodes 
of  Hertzian  waves  the  difference  of  potential  is  maximum, 
and  a  metal  bridge  placed  at  these  points  over  the  two  wires 
hinders  the  discharge  from  traversing  the  Geissler  tube. 
If  the  bridge  connects  the  wires  at  loops,  its  influence  is 
null  and  the  tube  lights  up.  Lecher  has  found  that  the  ve- 
locity of  propagation  of  Hertzian  waves  in  conductors  is 
close  to  the  velocity  of  light ;  Hertz  had  already  found  the 
same  number  for  the  velocity  in  air. 

Hitherto  we  have  only  considered  the  electric  field  whose 
mean  direction  coincides  with  that  of  the  axis  of  the  vibra- 
tor. But  the  periodic  currents  of  the  oscillating  discharge 
create,  in  addition,  a  magnetic  field  whose  lines  of  force  sur- 
round the  conductors  traversed  by  the  undulatory  electric 
flux(§  135). 

For  a  complete  investigation  of  the  phenomenon  we  must 
therefore  examine  at  the  same  time  both  the  effects  of  the 
electric  forces  propagated  in  form  of  waves,  and  of  the  per- 
pendicular magnetic  forces  which  accompany  these  disturb- 
ances in  the  surrounding  medium,  and  whose  effect  are 
added  to  the  first-mentioned.* 

*  For  the  details  of  Hertz's  experiments  see  :  Roosen,  Oscillations  ttectri- 


GENERAL    CONSIDERATIONS.  357 

224.  Present  Views  on  the  Propagation  of  Electric 
Energy. — Hertz's  experiments  are  a  brilliant  confirmation 
of  the  views,  set  forth  by  Faraday  and  defined  by  Maxwell, 
concerning  the  part  played  .by  the  medium  across  which 
electric  energy  is  transmitted^ 

It  will  be  remembered  (§  109)  that  one  of  the  hypotheses 
presented  to  account  for  electric  discharge  in  conductors 
consists  in  supposing  the  latter  to  be  the  seat  of  a  displace- 
ment of  electricity  comparable  to  the  movements  of  fluid  in 
pipes.  Hence  the  expressions  electric  current,  flux  of  elec- 
tricity, and  the  various  images  borrowed  from  the  dynamic 
theory  of  fluids  in  order  to  render  the  phenomena  more 
easily  understood  at  the  outset. 

But  the  investigation  of  the  properties  of  the  electric  cur- 
rents shows  that  there  is  a  profound  difference  between  it 
and  fluxes  of  ponderable  matter,  in  spite  of  the  analogies 
met  with  in  the  laws  governing  these  fluxes  (§  217). 

When  a  fluid  current  circulates  in  a  pipe,  no  external 
effect  shows  its  presence  ;  the  phenomenon  is  entirely  con- 
centrated inside  the  pipe  itself. 

The  electric  current,  on  the  contrary,  which  makes  itself 
evident  in  a  conductor  by  giving  out  heat,  exercises  peculiar 
and  very  striking  effects  in  the  surrounding  medium.  It 
magnetizes  the  medium,  as  is  shown  by  iron-filings  figures ; 
it  modifies  the  optic  properties  of  bodies  (§  167),  and,  lastly, 
it  produces  induction-phenomena  in  conductors  displaced  in 
its  field. 

Bends  in  a  pipe  cause  a  loss  of  kinetic  energy  and  dimin- 
ish the  blow  given  at  the  moment  of  stopping  the  flow  ;  the 
twisting  of  a  conductor  into  spirals  or  a  solenoid,  on  the 

ques  (Bull.  Assoc.  ing.  sort,  de  1'Inst.  Mont.,  1890);  PoincarS,  Electricitt  et 
Optique,  Paris,  Carr6,  1891  ;  Lodge,  The  Work  of  Hertz  and  some  of  his 
Successors,  Lond.,  1894. 


35$  THE  PROPAGATION  OF  CURRENTS. 

other  hand,  increases  the  energy  of  the  extra-current  on 
breaking  the  circuit. 

A  fluid  current  may  be  alternating ;  this  is  the  case  in  a 
pipe  where  the  sound-waves  are  propagated  in  the  form  of 
longitudinal  vibrations.  We  are  tempted  to  compare  such  a 
movement  to  an  alternating  electric  current ;  but  here,  too, 
differences  are  evident,  not  only  in  the  surrounding  space, 
but  even  inside  the  conductor  in  which  such  currents  pass. 
The  sound-wave  presents  maximum  displacements  along  the 
axis  of  the  pipe,  while  alternating  currents  have  their  greatest 
strength  towards  the  surface  of  the  conductor  (§  208). 

An  electric  current  must  be  considered  as  the  centre  of  a 
disturbance  which  affects  all  or  part  of  the  conductor  by  the 
Joule  effect,  and  which  extends  out  into  the  surrounding 
medium.  As  this  propagation  takes  place  in  a  vacuum,  it 
follows  that  it  is  the  ether  that  serves  as  vehicle  for  electric 
waves. 

A  current,  at  the  moment  it  starts,  excites  an  electromag- 
netic wave  which  is  transmitted  in  the  space  surrounding 
the  conductor  with  a  velocity  equal  to  that  of  light.  When 
the  current  has  reached  its  full  strength,  i.e.,  when  it  has  ac- 
quired a  constant  value  in  all  the  points  of  a  section  of  the 
conductor,  the  surrounding  medium  is  in  a  state  of  tension 
which  is  manifested  by  a  tendency  to  contract  in  the  direc- 
tion of  the  magnetic  lines  of  force  and  to  dilate  in  a  direction 
normal  to  them. 

The  ether  about  the  conductor  is  then  in  a  state  of  equi- 
librium characterized  by  cylindrical  layers  under  a  stress 
concentric  to  the  conductor.  When  the  current  ceases,  the 
ether,  being  suddenly  un-stressed,  falls  back  on  the  conduc- 
tor, giving  up  to  it  its  potential  energy,  which  is  shown  in 
the  extra-current. 

An  alternating  current  excites  continuous  waves,  which  as 
in  the  preceding  case  are  propagated  in  space  like  luminous 


GENERAL    CONSIDERATIONS.  3 $9 

waves ;  the  only  difference  lies  in  the  duration  of  the  period 
of  the  ether-vibrations. 

It  will  be  seen  further  on  that  alternating-current  dynamos 
give  from  50  to  200  vibratkms  per  second.  Given  the  enor- 
mous velocity  of  propagation  in  the  ether,  these  currents 
produce  waves  several  hundreds  of  kilometres  long.  The 
luminous  vibrations  of  a  lamp's  flame  amount  to  about  50 
trillions  per  second,  so  that  their  wave-length  is  only  some 
hundred-thousandths  of  a  centimetre. 

When  these  luminous  radiations  strike  a  body  which  in- 
tercepts them,  it  is  found  that  their  absorption  causes  a  de- 
velopment of  heat.  So,  too,  conducting  bodies  which  stop 
electric  radiations  are  the  seat  of  induced  currents,  made 
evident  by  calorific  phenomena.  The  induced  flux  of  elec- 
tricity is  in  the  same  direction  as  the  electric  force  and  nor- 
mal to  the  magnetic  force  of  the  wave. 

This  latter  penetrates  more  or  less  into  conductors, 
according  as  its  length  is  greater  or  smaller.  Short  electro- 
magnetic waves  only  affect  the  external  layers  of  the  con- 
ductors on  which  they  fall.  Hence  the  necessity  of  modify- 
ing the  conductor's  form,  and  adopting  for  short-period  cur- 
rents tubes,  strips  or  ropes  of  metal  in  preference  to  solid 
conductors. 

Electromagnetic  waves,  like  radiations  of  heat  and  light, 
necessarily  carry  off  with  them  a  part  of  the  energy  from  the 
source  they  leave  to  the  conductors  on  which  they  impinge. 
To  avoid  this  threatened  loss  in  alternating-current  circuits, 
the  circuit  is  made  of  an  outgoing  and  a  return  wire  placed 
very  close  together.  The  waves  emitted  by  the  first  go 
directly  to  the  second,  to  which  they  restore,  in  the  form  of 
induced  currents,  the  energy  radiated  off.  This  result  is 
most  completely  obtained  when  two  concentric  conductors 
are  used,  a  wire  and  a  tube  for  example,  for  then  the  circuit 
has  no  action  upon  a  neighboring  magnet,  and  the  magnetic 


360  THE  PROPAGATION  OF  CURRENTS. 

field  is  rigorously  limited  to  the  space  occupied  by  the  con- 
ductors and  the  dielectric. 

The  discovery  of  the  propagation  of  electromagnetic 
actions  in  the  form  of  waves,  similar  to  those  of  light,  is  of 
capital  importance.  It  establishes  an  intimate  bond  between 
electricity,  light,  and  heat,  and  will  doubtless  lead  to  im- 
portant progress  in  the  knowledge  of  the  laws  which  govern 
these  physical  agents. 

But  so  far  only  a  corner  has  been  lifted  of  the  veil  which 
hides  the  mechanism  of  the  transmission  of  electric  energy. 
We  have  learned  that  it  is  propagated  without  loss  in 
dielectrics,  while  conductors  are  the  seat  of  heat-effects 
which  entirely  or  partly  absorb  the  effective  energy.  But 
the  phenomenon  of  the  electric  current  still  remains  unex- 
plained, even  in  its  most  simple  form,  viz.,  after  the  variable 
period  is  over. 

From  all  modern  experiments  it  is  evident  that  an  electric 
current  is  the  manifestation  of  a  transfer  of  energy  which  is 
taking  place  in  the  medium  surrounding  the  conductors. 
These  latter  serve  only  to  direct  the  propagation,  and  they 
perform  this  part  at  the  expense  of  the  absorption,  under 
the  form  of  heat,  of  part  of  the  transmitted  energy.  A 
conductor  should  therefore  be  considered  as  the  directrix 
along  which  the  transfer  takes  place,  just  as  the  wick  of  a 
lamp  is  the  centre  of  the  flame  without  being  the  seat  of 
the  illuminating  effect.  The  seat  of  the  propagation  of 
electric  energy  is  in  the  electromagnetic  whirls  which  en- 
circle the  conductors.  As  to  the  real  mechanism  of  the 
transmission,  it  is  as  mysterious  as  the  mechanism  of 
gravitation.* 

*  Consult  :  Lodge,  Modern  Views  of  Electricity ;  Stoletow,  Ether  and 
Electricity  (The  Electrical  World,  Jan.  28  et  seq.,  1893). 


ELECTRICAL  MEASUREMENTS. 

MEASUREMENT  OF   ELECTRIC   POWER. 

225.  Case    of  a    Continuous  Current. —  The    electric 
power  developed  in  a  conductor,  subjected  to  a  potential- 
difference  e  and  traversed  by  a  current  z,  is  expressed  by 
the  product  ie. 

This  product  can  be  determined  indirectly  by  measuring 
the  factors  e  and  i  separately,  by  means  of  one  of  the 
methods  already  given,  or  directly,  by  means  of  instruments 
called  wattmeters. 

226.  Siemens  Wattmeter.  —  This  apparatus,  which    is 
applicable  to  the  measurement  of  power  developed  by  a 
continuous  current,  is  made  like  the  electrodynamometer, 
§152,  except  that  the  circuits  of  the  two  coils  are  separate. 
The  fixed  coil,  formed  of  a  great  number  of  turns  of  fine 
wire,  is  placed  in  shunt  to  the  conductor  in  which  the  power 
to  be  measured  is  developed  ;  the  moving  coil,  comprising  a 
few  turns  of  thick  wire,  is  in  series  with  this  conductor  and 
traversed  by  the  same  current  as  it. 

The  mutual  action  of  the  two  coils  is  proportional  to  the 
product  of  the  currents  which  traverse  them  ;  but  in  con- 
sequence of  the  high  resistance  of  the  stationary  coil  it  may 
be  assumed,  as  with  voltmeters,  that  the  current  traversing 
it  is  proportional  to  the  original  potential-difference  e  of 
the  ends  of  the  conductor.  The  electrodynamic  couple, 
measured  by  the  torsion-angle  8  through  which  the  suspen- 

361 


362  ELECTRICAL   MEASUREMENTS. 

sion-spring  must  be  turned  in  order  to  bring  the  movable 
coil  back  to  its  initial  position,  represents  very  nearly, 
therefore,  the  product  ei\  consequently 

e  =  kei. 

The  factor  k  is  determined  by  connecting  the  apparatus 
to  a  conductor  of  known  resistance  r,  traversed  by  a  cur- 
rent of  known  strength,  a  being  then  the  angle  of  torsion, 


227.  Case  of  a  Periodic  Current.  —  When   the  electric 
energy  is  transmitted  in  the  form  of  periodic  currents,  the 
choice  of  a  method  of  measuring  the  power  demands  special 
attention. 

There  are  then  two  cases  to  be  considered  : 

I.  The  conductor  in  which  is  developed  the  power  to  be 
measured  has  a  negligible   self-induction  ;    this  is  the  case 
with    straight    or   zigzagged    wires    and    coils   with    special 
windings. 

II,  The  self-induction  is  not  negligible. 

228.  Non-inductive  Conductors.  —  When  a  conductor  of 
the   first   of   the   above   classes   is   subjected  to   a  periodic 
potential-difference,  it  becomes  the  seat  of  a  current  whose 
phases  coincide  with  those  of  the  potential-difference.     Con- 
sequently, where  the  connection  between  the  current  and 
the  time  is  a  simple  sine-curve,  the  mean  power  is  equal  to 
the  product  of  the  effective  difference  of  potential  by  the 
effective  current  ;  for  we  then  have,  §  198, 


sin    -— 
T 


The   effective   current  can   be  determined    separately  by 
means  of  an  electrodynamometer,  or  by  a  voltmeter  having 


MEASUREMENT   OF  ELECTRIC  POWER.  363 

a  negligible  coefficient  of  self-induction — the  Cardew  volt- 
meter, for  example. 

If  we  have  to  deal  with  a  complex  periodic  function,  or 
one  the  form  of  which  is  unknown,  the  preceding  method 
should  not  be  used. 

On  the  hypothesis  that  the  power  under  consideration  is 
completely  transformed  into  heat  by  the  Joule  effect,  we 
can  then  deduce  the  quantity  of  this  heat  from  the  current 
or  potential-difference  given  respectively  by  the  readings  of 
an  electrodynamometer  or  a  Cardew  voltmeter.  Denoting 
by  r  the  resistance,  supposed  to  be  known,  of  the  conductor, 
we  have 

1  fat  =  rl  f?dt  =  - 
T t/o  T *A)  r 

The  Siemens  wattmeter  is  not  applicable  in  the  case  of 
periodic  E.  M.  Fs.  on  account  of  the  considerable  self- 
induction  of  the  fixed  coil.  The  current  produced  in  this 
coil  would  depend  on  its  impedance,  that  is,  on  the  fre- 
quency of  the  periods.  Zipernowski  has  sought  to  escape 
this  difficulty  by  making  each  of  the  coils  of  a  small  number 
of  turns,  so  as  to  render  their  self-induction  feeble.  At  the 
end  of  the  coil  which  is  to  be  placed  in  shunt  are  placed 
supplementary  coils,  wound  double,  whose  only  function  is 
to  supply  the  total  resistance  desired  in  the  shunt.  Call 
this  resistance  p,  r  that  of  the  conductor  in  which  we  are 
trying  to  find  the  power  developed,  and  R  the  resistance  of 
the  wattmeter's  second  circuit,  The  connections  of  the  ap- 
paratus are  generally  arranged  so  that  the  resistance  p  is 
in  parallel  to  the  two  resistances  r  and  R  placed  in  series. 

If  the  self-induction  Ls  of  the  branch  p  were  zero,  or,  more 

exactly,  if  the  time-constant  — -  were  negligible,  the  current 
traversing  this  branch  would  be  consonant  in  phase  with 


364  ELECTRICAL   MEASUREMENTS. 

the  potential-difference  e,  and  the  reading  of  the  apparatus 
would  be  directly  proportional  to  the  mean  power 


T     rT 
=  £-1  /  Mt, 

1  «/0 


the  constant  k  being  determined  in  the  same  manner  as  in 
the  Siemens  wattmeter. 

But  the  time  constant  —  of  the  shunt  is  not  negligible. 

The  result  of  this  is  a  diminishing  of  the  current  and  a  lag 
of  the  current-phase  behind  that  of  the  potential-difference, 
which  effects  cause  a  reduction  in  the  indications  of  the 
wattmeter  below  the  value  corresponding  to  the  power  actu- 
ally developed  in  the  conductor  r. 

As  Mather  has  shown,  the  indications  of  the  wattmeter 
may  be  rendered  correct  by  winding  the  coil,  which  is  put 
in  series  with  the  movable  coil,  in  such  a  way  that  its  ca- 
pacity neutralizes  the  effect  of  the  movable  coil's  self- 
induction. 

229.  Conductors  having  Self-induction. — The  preced- 
ing methods  are  still  at  fault  when  treating  a  conductor 
whose  self-induction  is  not  negligible. 

In  this  case  there  is  a  phase-lag  between  the  current  and 
the  periodic  E.  M.  F.  which  sets  it  up  in  the  conductor,  such 
that  we  have 


We  have  seen,  for  a  simple  periodic  function  that 

cos  0 


0  being  the  angle  of  lag. 

If  Zipernowski's  modified  wattmeter  is  applied  to  a  con- 
ductor having  self-induction,  there  is  produced  a  retardation 


MEASUREMENT  OF  ELECTRIC  POWER. 


365 


of  phase  both  in  the  main  circuit  of  the  apparatus,  in  series 
with  the  conductor,  and  in  the  shunt.  The  result  of  this 
is  a  tendency  towards  consonance  of  phase  in  the  two  cir- 
cuits, which  may  have  th/e  "effect  of  raising  the  readings  of 
the  instrument  beyond  the^value  corresponding  to  the  real 
power,  contrary  to  what  happens  in  the  case  of  a  conductor 
without  self-induction. 

This  error  is  avoided  if  Mather's  suggestion  is  utilized. 

In  the  case  under  consideration  we  can  arrive  at  rigorous 
results  by  the  use  of  the  calorimeter. 

Fig.  104  gives  a  general  idea  of  the  calorimeter  used  by 
M.  Roiti.  The  conductor  TT,  in  which  the  power  to  be 


FIG.  104. 

measured  is  developed  and,  by  hypothesis,  entirely  trans- 
formed into  heat,  is  enclosed  in  a  brass  vessel  placed  inside 
another  vessel  of  the  same  metal:  connecting  wires  enter 
through  tubes.  The  space  between  the  two  vessels  is  trav- 
ersed by  a  current  of  water  which  circulates  under  constant 
pressure  and  empties  into  a  gauged  receptacle  R.  Ther- 
mometers indicate,  within  a  tenth  of  a  degree,  the  tempera- 
ture of  the  water  on  entering  and  leaving  the  double 
covering  around  the  conductor.  When  the  temperatures 
have  become  constant,  which  is  sometimes  a  matter  of  many 


366  ELECTRICAL   MEASUREMENTS. 

hours,  the  number  of  grammes  of  water  flowing  through  per 
second  and  the  difference  in  reading  of  the  two  thermome- 
ters are  noted.  The  product  of  these  quantities  multiplied 
by  the  mechanical  equivalent  of  the  gramme-degree  is  the 
power  sought. 

Messrs.  Ayrton  and  Sumpner  have  worked  out  a  method 
which  allows  of  the  exact  determination  of  the  energy 
consumed  in  an  inductive  resistance  R,  by  means  of  a  volt- 
meter and  ammeter  for  alternating  currents.  A  non-induc- 
tor resistance  r  is  put  in  series  with  R,  causing  a  fall  of 
potential  comparable  with  that  caused  by  R.  Let  ul  be  the 
instantaneous  potential-difference  at  the  ends  of  R,  &2  that 
at  the  ends  of  r.  The  total  fall  of  potential  caused  by 
R  -\-  r  will  be  at  the  given  moment 

u=u,  +  ut  ........     (i) 

If  a  is  the  current  at  the  time,  the  power  at  the  given  in- 
stant is 

p  =  aul  =  —  u^. 

But  from  (i)  we  deduce 

*.*.  =  K«*  -  u'  ~  u?\ 
whence 

/  =  £(«'-«.'-<x 

The  mean  power  absorbed  by  R  will  be 

p  =  l 


where  £/,  U,,  U^  represent  the  effective  potential-differences 
at  the  ends  of  R-\-  r,  R  and  r  measured  with  a  Cardew  volt- 
meter or  an  electrometer.  If  r  is  not  known,  it  will  suffice 
if  the  effective  current  A  is  found  out  by  means  of  an  elec- 


MEASUREMENT   OF  A    MAGNETIC  FIELD.  367 

trodynamometer.  We  will  then  have,  observing  that  A  —  —  % 


MEASUREMENT   OF^THE  INTENSITY  OF  A  MAGNETIC   FIELD. 

230.  Method  by  Oscillation.—  When  a  magnetic  field 
may  be  considered  uniform  in  the  space  in  which  a  magnet- 
needle  moves,  the  duration  of  a  complete  oscillation  is  ex- 
pressed by 


, 

3TIJC' 

being  the  moment  of  inertia  of  the  needle  and  3T13C 
the  product  of  the  magnetic  moment  of  the  needle  by  the 
intensity  of  the  field  in  which  it  moves.  It  is  of  course 
taken  for  granted  that  the  field  does  not  modify  the  mag- 
netic moment  of  the  needle,  that  is,  that  its  intensity  of 
magnetization  corresponds  to  a  greater  magnetizing  force 
than  that  of  the  field. 

The  above  formula  gives  a  simple  method,  either  of  com- 
paring the  intensity  in  different  points  of  a  field,  or  of  deter- 
mining an  intensity  in  absolute  value  (§  48). 

231.  Electromagnetic  Method.  —  The  author  has  worked 
out  an  apparatus  which  allows  of  measuring  the  intensity  of 
a  magnetic  field  by  utilizing  its  action  upon  a  conductor 
traversed  by  a  known  current.  The  instrument,  shown  in 
Fig.  105,  was  constructed  by  MM.  F.  Pescetto  and  P. 
Zunini  when  they  were  students  at  the  Institut  Montefiore. 
The  conductor^!,  movable  about  an  axis  O  and  balanced  by 
a  counterweight  P,  is  traversed  for  a  portion  /  of  its  length 
by  a  current  i  led  in  by  flexible  wires/",/7,  and  which  can 
be  measured  in  an  ammeter. 


368  ELECTRICAL   MEASUREMENTS. 

If  3C  is  the  component  of  the  field-intensity  normal  to  the 
plane  of  the  conductor's  displacement,  the  electromagnetic 
force  acting  upon  this  latter  is 


This  force  is  counterbalanced  by  the  elastic  reaction  of  a 
spring-tongue  R  fixed  on  the  movable  conductor,  and  which 
is  acted  on  by  a  micrometer-screw  V.  This  screw  is  pre- 


FIG.  105. 

viously  graduated  by  determining  the  weight  which  must  be 
applied  to  the  centre  of  the  conductor  /in  order  to  balance 
the  spring  in  its  various  states  of  tension  and  to  bring  the 
two  arms  A  and  B  into  parallelism.  In  consequence  of  the 
smallness  of  the  electromagnetic  reactions  this  apparatus  is 
only  suitable  for  measuring  powerful  fields,  such  as  those  in 
dynamo  air-gaps,  into  which  the  flattened  arms  of  the  instru- 
ment can  be  introduced. 

232.  Method  Based  on  Induction. — The  method  most 
generally  employed  is  that  described  in  §  192,  which  is 
based  on  induction.  This  method  enables  us  to  measure 
the  intensity  of  very  confined  fields,  such  as  that  of  a  dy- 
namo air-gap,  for  it  is  almost  always  possible  to  introduce 
a  small  flattened  coil,  connected  to  a  ballistic  galvanometer. 
Tf  the  coil  be  then  sharply  withdrawn  to  a  position  in  which 


MEASUREMENT  OF  MAGNETIC  PERMEABILITY.     369 

the  flux  of  force  traversing  it  is  negligible,  a  displacement  of 
electricity  is  produced  in  the  circuit,  which  is  measured  by 
the  angle  of  swing  a  of  the  galvanometer. 

We  have,  §  150,  > 

q  =  £.4rin  \OL. 

But 


whence 

kR  sin 


If  the  ballistic  constant  k  of  the  galvanometer  is  unknown, 
it  is  determined  by  discharging  into  the  apparatus  a  stand- 
ard condenser  charged  by  means  of  a  cell  of  known  E.  M.  F., 
or  by  introducing  into  the  galvanometer-circuit  an  inclinom- 
eter (§  192)  which  is  made  to  turn  in  the  earth's  field  so  as 
to  generate  a  calculable  flux  of  electricity.  * 

MEASUREMENT  OF   MAGNETIC  PERMEABILITY. 

233.  Methods  Based  on  Induction. — I.  The  apparatus 
shown  in  Fig.  106  has  been  used  by  Dr.  Hopkinson.  The 


FIG.  106. 

iron  bar,  whose  permeability  is  to  be  measured,  is  formed  of 
two  sections,  C  and  C' ;  it  passes  through  a  mass  of  wrought 
iron  A,  two  magnetizing  coils  BB,  and  a  small  coil  D  con- 
nected with  a  ballistic  galvanometer. 

This  coil  D  is  pulled  laterally  by  an  elastic  thread,  so  that 
if  the  two  sections  C  and  C'  be  slightly  separated  the  coi]  is 


37°  ELECTRICAL   MEASUREMENTS. 

drawn  out  of  the  apparatus  and  traversed  by  an  electric  flux 
in  proportion  to  the  magnetic  induction  of  the  bar. 

The  iron  mass  and  the  bar  form  a  magnetic  circuit  trav- 
ersed by  a  flux  produced  by  the  magnetomotive  force  ^nnc, 
n  and  c  being  respectively  the  total  number  of  turns  and 
the  current  of  the  coils  BB. 

Let  #  be  the  flux,  /  that  length  of  the  bar  inside  the  open 
space  in  A,  s  its  section,  /*  its  permeability;  also,  let  I'  be 
the  mean  path  of  the  lines  of  magnetic  force  in  the  mass 
A,  s'  its  section,  X  its  permeability. 

We  have  (§  164) 


In  commercial  tests,  where  a  very  close  approximation  is 
not'  called  for,  the  second  term  of  the  binomial  may  be 
neglected  in  comparison  with  the  first,  and  we  have  then 
simply 

.       <£/ 
4***  =  —  .......     (i) 

Now  if  nf  denote  the  number  of  turns  in  the  coil  D,  the 
electric  flux  produced  by  withdrawing  this  coil  is 

ay 

q  =  -£-  =  k  sin  i«,       .....     (2) 

R  being  the  electric  resistance  of  the  circuit  comprising  the 
ballistic  galvanometer. 

By  varying  the  magnetizing  current  c,  measured  in  an  am- 
meter, we  can  deduce  from  equations  (i)  and  (2)  the  values 
of  the  permeability  corresponding  to  different  values  of  the 
magnetizing  force.  It  is  by  means  of  such  experiments  that 
Dr.  Hopkinson  has  been  enabled  to  draw  the  continuous 
curves  of  Fig.  22,  in  which  the  permeability  of  various 
samples  of  iron  is  represented  in  terms  of  the  magnetic  in- 
duction. 


MEASUREMENT  OF  MAGNETIC  PERMEABILITY.     371 

234. — II.  Fig.  107  shows  an  arrangement  but  slightly  dif- 
ferent from  the  preceding.  The  wrought-iron  mass  is  not 
shown  for  the  sake  of  clearness  in  the  figure. 

The  bar  B,  surrounded  by  the  mass  of  wrought  iron,  is  un- 
divided and  fixed v  It  parses  through  the  magnetizing  coil 
M,  traversed  by  a  current  measured  in  the  ammeter  a,  and 
the  coil  5  connected  to  the  ballistic  galvanometer  G.  This 


FIG.  107. 

latter  is  graduated  beforehand  by  means  of  a  standard  con- 
denser C  and  a  standard  cell  E,  which  enable  us  to  deter- 
mine the  ballistic  constant  k. 

When  the  direction  of  the  current  in  M  is  changed  by  the 
key  C't  the  flux  which  traverses  the  turns  of  the  coil  5  varies 
by  2$.  Hence  there  is  an  electric  flux  in  the  galvanometer 

2$n'       ,    .    a 


=-. 


.  . 
(i) 


The  rheostat  R'  serves  to  adjust  the  total  resistance  R  so 
as  to  obtain  suitable  swings  of  the  needle  and  to  prevent  the 
galvanometer  from  becoming  dead-beat  in  the  case  where  a 
Deprez-d'Arsonval  instrument  is  used. 

If,  instead  of  reversing  the  magnetizing  current,  it  is  cut 
off,  there  remains  a  flux  in  the  bar  due  to  the  residual  mag- 
netism of  the  system  formed  by  the  core  and  the  envelop- 
ing mass.  The  electric  flux  q'  measures  in  this  case  the 
difference  between  the  total  magnetic  flux  and  the  residual. 

By  adding  to  equation  (i)  the  relation 

$~,  .......     (2) 


372  ELECTRICAL   MEASUREMENTS. 

we  can  deduce  the  values  of  yw  from  the  results  of  experi- 
ments. 

To  determine  precisely  the  residual  magnetism  of  the 
metal  to  be  tested,  Prof.  Rowland  gives  the  test-piece  the 
form  of  a  ring,  the  thickness  of  which  along  the  axis  is  very 
much  greater  than  the  thickness  normally  to  the  axis.  On 
completely  surrounding  such  a  ring  with  a  magnetizing  coil, 
a  constant  field  is  obtained  within  it,  of  intensity  OC  =  ^Ttnj 
(§  153).  An  auxiliary  coil,  wound  round  the  ring  and  con- 
nected to  a  ballistic  galvanometer,  enables  the  total  flux 
across  the  core  to  be  measured.  By  this  arrangement  there 
is  no  need  of  an  additional  iron  mass,  and  the  residual  mag- 
netism of  the  core  can  be  measured  exactly ;  but  the  prepa- 
ration and  winding  present  difficulties. 

235.  In  order  to  obtain  by  the  induction-method  the  suc- 
cessive points  of  the  curve  representing  the  increases  of 
intensity  of  magnetization  corresponding  to  increases  in  the 
magnetizing  force,  Fig.  18,  it  is  indispensable  not  to  proceed 
by  reversals  or  interruptions,  but  by  gradual  increments  or 
decrements  of  the  current  in  the  magnetizing  coil.  The  cor- 
responding impulses  observed  in  the  ballistic  galvanometer 
enable  us  to  calculate  the  progressive  increases  of  magnetic 
flux  across  the  core  and,  consequently,  the  values  of  the  mag- 
netic induction  and  intensity  of  magnetization.  After  find- 
ing by  this  means  the  points  corresponding  to  a  complete 
cycle  of  the  magnetizing  force,  we  can  determine  the  loss  by 
hysteresis  in  the  cycle.  This  method  has  the  disadvantage 
of  accumulating  in  the  curves  the  successive  errors  committed 
in  determining  the  separate  points. 

It  would  be  possible  to  avoid  this  accumulation  of  errors 
by  substituting  a  straight  cylindrical  core  for  the  shapes 
proposed  by  Messrs.  Hopkinson  and  Rowland,  on  condition 
of  giving  the  bars  or  wires  a  length  equal  to  400  or  500  times 


MEASUREMENT  OF  MAGNETIC  PERMEABILITY.     3/3 

their  diameter,  so  that  the  magnetization  could  be  consid- 
ered uniform  in  the  medial  region  of  the  bar  when  intro- 
duced into  a  magnetizing  coil  of  similar  length  and  section 
(§  55)-  The  coil  connected  to  the  ballistic  galvanometer 
must  then  be  wound  outside  the  magnetizing  coil  and  put 
near  its  middle  portion.  If  this  test-coil  is  then  drawn 
sharply  back,  we  will  get  an  impulse  in  the  galvanometer 
enabling  us  to  calculate  the  total  flux  produced  by  the  core 
and  the  magnetizing  coil.  To  prevent  the  vibrations,  caused 
by  drawing  back  the  test-coil,  from  being  communicated  to 
the  core  and  affecting  its  magnetization,  this  coil  may  be 
wound  on  a  glass  tube  surrounding  the  magnetizing  coil. 

236.  Magnetometric  Method. — Instead  of  treating  the 
specimen-bar  by  the  induction-method,  its  magnetic  moment, 
plus  that  of  the  magnetizing  coil,  can  be  directly  determined 
by  the  magnetometric  method  described  in  §  48.  For  this 
purpose  the  bar  is  placed  in  the  plane  of  a  suspended  mag- 
net-needle, and  by  the  deviation  of  the  needle  we  calculate 
the  ratio  of  the  total  moment  sought  to  the  horizontal  com- 
ponent of  the  earth's  field.  This  latter,  and  also  the  effect 
of  the  magnetizing  coil  on  the  needle,  are  separately  deter- 
mined. For  a  straight  electromagnet  having  the  proportional 
dimensions  indicated  above,  the  poles  are  practically  situated 
at  the  ends  of  the  core.  The  electromagnet  may  be  placed 
vertically ;  and  in  this  case,  if  its  length  is  sufficient  and  if 
one  of  the  poles  is  very  close  to  the  needle,  the  effect  of  the 
other  pole  becomes  negligible. 

This  proceeding  gives  us  the  magnetic  moment  and,  con- 
sequently, the  intensity  of  magnetization  of  the  rod  for 
cyclic  values  of  the  magnetizing  force,  whence  we  deduce 
the  corresponding  loss  by  hysteresis.  The  magnetometric 
method  has  the  disadvantage  of  necessitating  the  determi- 
-  nation  of  the  horizontal  component  of  the  earth's  field. 


374 


ELECTRICAL   MEASUREMENTS. 


237.  Method  by  Portative  Power. — The  portative  power 
of  a  magnet  is  expressed  in  terms  of  its  magnetization  by 
the  formula 

/  =  27i$ s.     (§  56) 

If  the  magnet  be  under  the  influence  of  a  magnetic  field, 
we  must  add  a  second  term,  3CO^,  and  then  we  have 

p  —  27r32j -f  OCay (i) 


Now 
and 


(§52) 


//  =   I  -f  47T/C.       . 

Combining  (i),  (2),  and  (3),  we  obtain 


(2) 
(3) 


Now  for  a  very  long  solenoid  comprising  «x  turns  per  cm. 
of  length  and  traversed  by  a  current  i,  JC  is  sensibly  equal  to 


MM.  M6lotte  and  Henrard,  students  at  the  Liege  Electro- 
technical  Institute,  have  utilized   these   properties  in  the 


FIG.  108. 

construction  of  a  commercial  apparatus  for  measuring  per- 
meabilities. The  bar  to  be  tested  is  divided  into  two  parts, 
M,  F,  placed  in  the  middle  of  along  coil  completely  enclosed 


MEASUREMENT  OF  COEFFICIENTS   OF  INDUCTION.  375 

in  an  iron  sheath.  The  part  M  is  hung  from  one  of  the  ends 
of  a  balance-beam  A.  The  other  end  of  this  beam  carries  a 
vessel  P,  into  which  sand  is  run  through  a  tube  T  until  the 
two  parts  of  the  bar  are  separated,  against  the  magnetic  at- 
traction, produced  by  the  jgissage  of  a  current  in  B,  which 
tends  to  hold  them  in  contact. 

The  part  M  is  slightly  conical,  and  rests  by  its  smaller 
base  upon  the  larger  end  of  F;  in  this  way  a  well-defined 
surface  of  contact  is  obtained. 

This  method  cannot  give  the  values  of  the  permeability 
corresponding  to  small  values  of  the  magnetizing  force,  nor 
yet  the  necessary  data  for  calculating  the  loss  by  hysteresis ; 
but  for  the  values  of  permeability  met  with  in  dynamos,  the 
results  agree  practically  with  those  obtained  by  the  preced- 
ing methods. 

MEASUREMENT  OF  COEFFICIENTS   OF  INDUCTION. 

238.  Maxwell  and  Rayleigh's  Method  for  Measuring 
a  Coefficient  of  Self-induction  in  Terms  of  a  Resistance. 

—The  coil  d,  whose  coefficient  of  self-induction  is  Ls,  is  in- 
serted  in  the  fourth  arm  of  a  Wheatstone  bridge,  Fig.  109, 
set  up  with  a  ballistic  galvanometer  and  three  series  of  non- 


FIG.  109. 

inductive  resistances.  These  last  are  commonly  formed  of 
coils  with  double  windings  whose  self-induction  is  not  en- 
tirely negligible  and  which  also  possess  an  electrostatic 


ELECTRICAL   MEASUREMENTS. 

capacity  sufficient  to  cause  sensible  errors  in  the  measure- 
ment of  feeble  coefficients  of  self-induction.  In  such  a  case 
it  is  much  better  to  employ  very  slender  wires,  straight  or 
in  a  zigzag. 

The  four  arms  are  first  adjusted  in  the  ordinary  wa)'  on  a 
constant  current ;  next,  the  galvanometer  arm  is  closed 
before  that  containing  the  battery ;  the  reactions  due  to 
self-induction  of  the  coil  produce  a  deviation  a.  When  the 
battery  circuit  is  opened,  there  is  an  equal  and  contrary 
deviation. 

Denoting  by  Dl  the  current,  when  fully  established,  in  the 
arm  d,  the  quantity  of  electricity  conveyed  by  the  extra- 
current  on  opening  is  (§  194) 


0  = 


the  portion  of  this  flux  flowing  through  the  galvanometer  is 

=  £  sin  £a,    .     (l) 


where  k  denotes  the  ballistic  constant  of  the  galvanometer 
(§  ISO). 

Lastly,  the  ohmic  equilibrium  of  the  bridge  is  destroyed 
by  adding  to  d  a  small  resistance  r,  which  produces  a  per- 
manent deflection  d  in  the  galvanometer. 

The  current  in  the  galvanometer  is  then  approximately 
the  same  as  if  an  electromotive  force  equal  to  rZ?a  were 
acting  in  the  arm  d,  the  line  containing  the  battery  being 
interrupted.  We  then  have 

rD  °  +  i 


t  \ 

a  + 
K  being  the  reduction-factor  of  the  galvanometer. 


MEASUREMENT  OF  COEFFICIENTS  OF  INDUCTION.  ^77 

By  combining  equations  (i)  and  (2)  we  deduce 


f^y, 

| 
When  a  sensitive  galvanometer  is  employed,  the  resistance 

to  be  added  to  d  is  very  small,  so  that  the  currents  Dl  and 
Z>a  differ  only  by  a  negligible  quantity,  and  their  ratio  is 
practically  equal  to  unity ;  the  formula  then  reduces  to 

k  sin  l>a 


The  constants  k  and  K  are  determined  by  preliminary 
experiments. 

239.  Maxwell's  Method,  modified  by  Pirani,  for  Meas- 
uring a  Self-induction  in  Terms  of  a  Capacity. — The 

relation  existing  between  the  effects  of  a  self-induction 
and  a  capacity  in  a  circuit  traversed  by  a  variable  current 
(§  219  et  seq.)  has  suggested  a  number  of  methods  for  meas- 
uring one  of  these  quantities  in  terms  of  the  other.  The 
following  one  is  especially  convenient. 

A  Wheatstone  bridge  (Fig.  no)  has  three  non-inductive 
arms,  a,  b,  s.  In  the  fourth  arm  is  placed  the  coil  ry  whose 
coefficient  of  self-induction  is  sought,  in  series  with  a  non- 
inductive  resistance  r' .  A  condenser  of  capacity  C  is  put 
in  shunt  on  this  latter. 

First  of  all,  the  four  arms  are  balanced  for  a  fully  estab- 
lished current;  then  they  are  balanced  during  the  variable 
period  by  varying,  for  example,  the  point  of  connection  of 
the  condenser  to  the  non-inductive  resistance.  Let  R  be  the 
resistance  of  this  fraction  when  the  galvanometer  remains 
at  zero  with  the  battery-circuit  either  open  or  closed.  It 
was  shown  in  §  219  that,  in  regard  to  the  flux  of  electricity 


378 


ELECTRICAL   MEASUREMENTS. 


transmitted  during  the  variable  period,  the  condenser-effect 
corresponds  to  a  decrease  of  self-induction  equal  to  the  pro- 
duct of  the  capacity  by  the  square  of  the  resistance  of  the 
conductor  in  parallel  with  the  condenser. 


We  shall  then  have  in  the  present  case 

A  =  kR\ 
where  k  is  the  capacity  of  the  condenser. 

240.  Ayrton  and  Perry's  Method  for  Comparing  a  Coil's 
Self-induction  with  that  of  a  Standard  Coil. — Ayrton  and 
Perry's  standard  of  self-induction  enables  the  self-induction 
of  a  coil  to  be  determined  in  the  quickest  possible  way. 


i > 


FIG.  in. 


For  this  purpose  a  bridge  is  made  (Fig.  1 1 1),  by  means  of  two 
non-inductive  arms  b,  d,  the  coil  to  be  standardized  c,  and 
the  standard  a.  The  galvanometer  is  brought  to  zero  for  a 


MEASUREMENT  OF  COEFFICIENTS  OF  INDUCTION.  379 

fully  established  current,  then  the  battery-circuit  is  opened 
and  closed  while  altering  the  position  of  the  movable  coil  in 
the  standard  until  the  galvanometer  remains  at  zero  during 
the  variable  period.  We  ,then  have  the  proportion 


~  —       "a 


This  method  does  not  necessitate  the  long  series  of  small 
adjustments  needed  by  the  other  methods  in  order  to  ob- 
tain a  balance  in  the  two  states  (fully  established  and  varia- 
ble) of  the  current.  If  the  self-induction  of  the  resistance  c 
is  beyond  the  bounds  of  the  standard,  it  is  only  necessary  to 
modify  the  ratio  of  the  arms  b  and  d,  as  is  done  in  compar- 
ing, by  Wheatstone's  bridge,  very  different  resistances. 

Messrs.  Ayrton  and  Perry  have,  moreover,  designed  a 
commutator  which  markedly  increases  the  sensitiveness  of 
both  this  and  the  preceding  methods,  by  accumulating  the 
effects  on  the  galvanometer  of  the  impulses  due  to  repeated 
making  and  breaking  of  the  battery-circuit,  For  this  pur- 
pose the  connections  of  the  bridge  with  the  galvanometer 
and  the  battery  are  made  by  means  of  metallic  brushes  bear- 
ing on  two  movable  discs  B,  R,  composed  of  annular  con- 
ducting segments  separated  by  insulating  spaces.  The 
movements  of  these  discs  are  interdependent  and  adjusted 
so  that  the  battery-circuit  may  be  alternately  open  and 
closed,  while  the  galvanometer  is  successively  connected  to 
the  bridge  and  short-circuited.  The  periods  of  connection 
of  the  galvanometer  to  the  bridge  correspond  to  the  variable 
periods  of  opening  and  closing,  and  the  connections  are 
alternated  in  such  a  manner  that  the  momentary  currents 
traverse  the  galvanometer-coil  in  the  same  direction.  With 
this  cumulative  method  a  slight  error  in  equilibrium  is 
shown  by  a  perceptible  deflection  of  the  needle. 


380  ELECTRICAL   MEASUREMENTS 

Note.  —  When  investigating  an  electromagnet  by  the 
preceding  methods,  the  results  found  for  the  self-induction 
vary  with  the  current  which  traverses  the  field-coils,  since 
the  coefficient  of  self-induction  is  a  function  of  the  permea- 
bility of  the  core  (§  188).  It  is  therefore  necessary  to  indi- 
cate the  maximum  value  of  the  current  traversing  the  field- 
coils  for  each  value  of  the  coefficient  of  self-induction. 

241.  Mutual  Induction—  Carey-Foster  Method.—  Sup- 
pose two  concentric  coils  with  or  without  an  iron  core  ;  de- 
note by  Lm  their  mutual  induction  ;  insert  one  of  them  in  a 
circuit  through  which  a  constant  current  i  is  sent  ;  a  flux  of 
magnetic  force  Lmi  then  traverses  the  second  coil.  If  this 
latter  is  in  communication  with  a  ballistic  galvanometer  so 
as  to  form  a  circuit  of  total  resistance  R,  the  galvanometer 
is  traversed  by  a  flux  of  electricity,  measured  by  a  deflection 
or,  when  the  current  in  the  first  coil  is  stopped.  We  have 


(i) 


A  condenser  of  capacity  c  is  next  charged  by  connecting 
its  plates  to  the  extremities  of  a  resistance  traversed  by  the 
same  current  i.  The  discharge  of  the  condenser  into  the 
same  galvanometer  gives  a  deflection  a',  such  that 


(2) 
Equations  (i)  and  (2)  give 


sn 


Carey-Foster  has  given  an  arrangement  enabling  the  two 
preceding  combinations  to  be  performed  simultaneously  so 
as  to  balance  the  effects  of  the  two  electric  fluxes  on  the 


MEASUREMENT  OF  COEFFICIENTS  OF  INDUCTION.  381 

needle.  The  galvanometer  G,  Fig.  112,  is  connected  to  the 
condenser  and  to  the  resistance  r,  on  the  one  hand,  and  to 
the  circuit  of  resistance  R  on  the  other.  By  varying  r  until 


S^  5 

l..«K.nAftAA»\nA/N(WU-— — i  $ 

P],|,|L_    L*__M^J 


FIG.  112. 

the  needle  remains  at  zero,  both  on  making  and  breaking 
the  battery-circuit,  we  have  the  relation 


Lm  =  crR. 


INDEX. 

NUMBERS  REFER  TO  ARTICLES,  AND  NOT  TO  PAGES. 


Absolute  electrometer,  96 
Action  at  a  distance,  9 

"       of  a  magnetic  field  on  an  element 

of  current,  138 

"        n  ic  uniform  field  on  a  magnet,  37 
"        "  earth  on  a  magnet,  33 
"        "  homogeneous  sphere  on  external 

point,  28 
"        "  homogeneous  sphere  on  internal 

point,  291 
"        "  spherical  shell  on  external  point, 

27 
*«        "  spherical  shell  on  internal  point, 

26 

Actions,  Law  of  electric,  82 
Alternating  current,  Method  for  determin- 
ing hysteresis,  69 

Alternating  currents.  Combined  effects  of  a 
capacity  and  a  self-induction  in  a  cir- 
cuit traversed  by, 221 
Alternating  currents,  Effect  of  capacity  in 

a  circuit  traversed  by,  220 
Alternating  currents,   Special  characteris- 
tics shown  by  (Mode  of  combining),  218 
Ampere's  hypothesis  on  the  nature  of  mag- 
netism, 141 

Angles,  Measurements  of,  in  radians,  50 
Annular  coil,  153 

"        magnet,  54 
Application  of  theorems  relative  to  central 

forces,  26  to  31 
Arago's  disk,  209 
Artificial  magnets,  47 
Attraction,  Law  of  magnetic,  34 
Ayrton  and  Perry's  method  for  comparing 
self-induction  of  coils,  240 


Ballistic  method  for  determining  hystere- 
sis, 69 

Barlow's  wheel,  157 
Becquerel's  law,  127 


Bobbin,  cylindrical,  151 

"        ring-shaped,  153 

Bodies  in  a  magnetic  field,  Modifications  of 
properties  of,  167 

"      Magnetic  and  diamagnetic,  51 
Body  in  a  magnetic  field,  Equilibrium  of,  65 
Bridge,  Wheatstone's,  123 
British  Association  unit,  180 
Brush-discharge,  Rotation  of,  by  a  magnet, 

156 
"       Electric,  no 

C 

Capacitance  and  inductance,  214 
Capacity  and  self-induction,  Combined  ef- 
fects of,  in  a  circuit  traversed  by  alter- 
nating currents,  221 
Capacity  and  self-induction  of  a  circuit, 

Comparative  effects  of  the,  219 
Capacity  (Electrostatic)  of  Conductors,  92 
"         Effect  of,  in  a  circuit  traversed 

by  alternating  currents,  220 
"         of  condenser  not    connected  to 

earth,  98 

Carey-Foster  method    for  mutual    induc- 
tion, 241 

Cell,  Voltaic,  129 

Central  forces,  general  theorems,  9  to  25 
"  "         definitions,  9 

"  "        elementary  law  governing, 

10 

C.  G.  S.  and  practical  systems,  Relation  be- 
tween, 177 

"         system  of  units,  171 
Charged  conductor  in  equilibrium,  81 
Chemical  effect  of  the  current,  127 
"          electromotive  forces,  113 
Circuit,  Coefficient  of  self-induction  of,  163 
"       composed    of    linear    conductors, 
Self-induction  in  a,  194 
Circuit    composed    of    linear    conductors. 
Self-induction  where  there  is  a  periodic 
or  undulatory  electromotive  force,  198 
383 


3^4 


INDEX. 


Circuit,  electric,  Magnetic  potential  due  to 
an,  141 

Circuit  electric,  Work  due  to  the  displace- 
ment of,  under  the  action  of  a  pole,  140 

Circuit,  Magnetic,  164 

"       traversed  by  a  current,  Reactions 
produced  in  a, 160 

Circular  electric  current,  Magnetic  poten- 
tial due  to  a,  147 

Closing  a  circuit,  Work  accomplished  on, 

'95 
Coefficient  k  in  Coulomb's  law,  Nature  of, 

104 

"          of  hysteresis,  71 
"          "  "          Table  of  values,  71 

"          "  self-induction  of  a  circuit,  163 
Coil,  Annular,  153 

"     Cylindrical,  151 

Combined  effects  of  a  capacity  and  a  self-in- 
duction in  a  circuit  traversed  by  alter- 
nating currents,  221 
Comparative  effects  of  the  self-induction 

and  capacity  of  a  circuit,  219 
Condenser,  Capacity    of,   when    not    con- 
nected to  earth,  98 
"  Cylindrical,  97 

"  Discharge  of,  into  galvanome- 

ter with  shunt,  197 
"  guard-ring,  95 

plate,  94 

"  Residual  discharge  of  a,  106 

"  Spherical,  93 

Conductive  discharge,  109 
Conductor,  cylindrical,   Self-induction    in 

the  mass  of  a,  208 
Magnetization  of,  by  a  current, 

1 66 
Conductors  and  insulators,  77 

"  Electrostatic  capacity  of,  92 

Congress,  electrical,  of  1893,  Recommenda- 
tions of,  181 

Conservation  of  energy;  application  to  elec- 
trolysis, 129 
"  "        "       Principle  of,  6 

"       matter,  6 

Constant  difference  of   potential  in  a  con- 
ductor, Means  of  keeping  up,  113 
Constant  electromotive  force,  Case  of,  194 
Construction  and  forms  of  electromagnets, 

165 

Contact  electromotive  force,  107 
Continuous  current,  Measurement  of  pow- 
er of,  225 

Convective  discharge,  to8 
Cores  of  electromagnets  traversed  by  vari- 
able currents.  207 


Corresponding  elements  of  tube  of  electric 

force,  88 
Coulomb's  law,  Nature  of  coefficient  k  in, 

104 

"     Proof  of,  49 
"          theorem,  86 

Current,  circular  electric,  Magnetic  poten- 
tial due  to  a,  147 
•'         continuous,     Measurement      of 

power  of,  225 
"         electric,  109 
"  "       Action    of    a  magnetic 

field  on  an  element  of,  138 
"         electric,  Magnetic  field  due  to  an 

infinite  rectilinear,  136 
44         electric,  in  a  magnetic  field,  En- 
ergy of,  142 

"        electric,  Intrinsic  energy  of,  144 
"  "         Rotation  produced  byre- 

versing,  158 

Current,  element  of.  Work  due  to  the  dis- 
placement of,  under  the  action   of  a 
pole,  139 
Currents,  infinite  rectilinear,  Potential  due 

to,  146 
Current,  mean  and  effective.  Measurement 

by  dynamometer,  200 
Current-meter,  Shallenberger's,  211 
Current  periodic,  power  of,  227 

"  "        power  of  inductive  con- 

ductors, 229 
"        power   of   non-inductive 

conductors,  228 
Current,  phenomena  w.^.ich  accompany  the 

propagation  of  in  a  conductor,  217 
Current,  Rotation  of,  by  a  magnet,  155 
"        Variable  period  of,  117 
"        Variable  period  of  application  of 
Ohm's  law  in  but  slightly  conductive 
bodies,  118 

Currents,  alternating,  Combined  effects  of 
a  capacity,  and  a  self-induction  on  a 
circuit  traversed  by,  221 
Currents,  alternating,  Effect  of  capacity  in 

a  circuit  traversed  by,  220 
Currents,  alternating,  Special  characterist- 
ics shown  by  (Mode  of  combining),  218 
Currents,  Eddy,  72 

"  "       effects  of,  73 

"        electric,  Induced,  182 

"  "         Mutual  action  of,  159 

"  "         Relative  energy  of  two, 

H3 

"        Foucault,  206 

"  "          Calculation  of  power 

lost  in,  207 


INDEX. 


385 


Currents  induced,  Rotation  due  to,  209 
Cylindrical  bobbin,  151 

"  condenser,  97 

"  conductor,    Self-induction    in 

the  mass  of,  208 

"  (infinite)  magnet,  55 


Definitions  of  magnetic  quantities,  36 
Density,  Surface  and  volume  (defined),  9 
Derived  circuits,  Application  of  Kirchhoff  s 

Laws  to,  122 
Determination  of  magnetic  moment  of  a 

magnet,  48 

Diamagnetic  and  magnetic  bodies,  51 
Dielectrics,  Specific  inductive  capacity  of, 

103 
Difference  of  potential   and  electromotive 

force,  Distinction  between,  107 
Difference   of    potential    in  a    conductor, 

Means  of  keeping  up,  113 
Dimensions  of  derived  unit,  4 
"  "   A' and  ju.,  175 

"  "   units,  170 

"  "      "       applications  of,  8 

"          Theory  of,  4 
Disc,  Arago's,  209 
"      Faraday's,  191 

"      Infinitely  thin  potential  due  to  an,  31 
"    -magnet,  53 
Discharge,  Conductive,  109 
"  Convective,  108 

"  Disruptive,  no 

"  Instantaneous    electric    meas- 

urement of  an,  150 

"          of  a  condenser  into  a  galvanom- 
eter with  shant,  197 
"  Oscillating,  222 

Discharging  power  of  electrified  points,  89 
Discovery,  Oersted's,  135 
Displacement,  Electric,  in  dielectrics,  105 
Disruptive  discharge,  no 
Distinction   between    electromotive  force 
and  difference  of  potential,  107 

E 

Earth,  Electric  potential  of  the,  85 
Eddy  currents,  72 

"  "          Effects  of,  73 

Effect,  Feranti,  221 
"       Hall,  168 
"       Joule's,  127,  196 
"       Kelvin,  131 

' '      of  capacity  in  a  circuit  traversed  by 
alternating  currents,  220 


Effect,  of  the  current,  Chemical,  127 
"      Peltier,  126,  130 
'•       Seebeck,  130 

Effects,    Combined,    of  a  capacity  and  a 
self-induction  in  a  circuit  traversed  by 
alternating  currents,  221 
Effects,  Comparative,  of  the  self-induction 

and  capacity  of  a  circuit,  219 
Electric  actions,  Law  of,  82 

"        circuit,  Magnetic  potential  due  to 

an,  141 

Electric  circuit,  Work  due  to  the  displace- 
ment of,under  the  action  of  a  pole, 140 
"       conductor,  Magnetization  of,  by  a 

current,  166 
"       current,  109 
"  u        Action  of  a  Magnetic  field 

on  an  element  of,  138 
"  "        Chemical  effect  of,  127 

"  "        circular,  Magnetic  poten- 

tial due  to  an,  147 

Electric  current,  element  of,  Work  due 
to  the  displacement  of,  under  the  action 
of  a  pole,  139 

Electric    current,  Energy   of,  general  ex- 
pression, 124 

"  "          Energy  of    heterogene- 

ous conductor,  126 

"  "          Energy    of    homogene- 

ous conductor,  125 
"  "          in    a     magnetic     field, 

Energy  of,  142 

Electric  current,  infinite  rectilinear,  Mag- 
netic potential  due  to  an,  146 
Electric  current,  Intrinsic  Energy  of,  144 
"  "         Laws     of      (preliminary 

note),  in 
"  "         Magnetic  field  due  to  an 

infinite  rectilinear,  136 
Electric    current,     mean     and    effective, 
Measurement  of,  by  electro-dynamom- 
eter, 200 

Electric    current,    phenomena    which    ac- 
company the  propagation  of,  in  a  con- 
ductor, 217 
Electric  current,  Rotation  of,  by  a  mag 

net,  155 
"  "         Rotation    produced    by 

reversing,  158 

"        currents,  Induced,  182 
"  "  Mutual  action  of,  159 

u  "          Relative  energy  of  two, 

i43 

Electric  discharge,  instantaneous,    Meas- 
urement of,  150 
"        displacement  in  dielectrics,  105 


386 


INDEX. 


Electric    energy,   Present    views   of   the 

propagation  of,  224 
"        field  (defined),  83 
44         potential  (defined),  83 

"         of  the  earth,  85 
41        screen,  go 
"         spark  and  brush,  11° 
"        waves,  transmission  of,  in  the  sur- 
rounding medium,  223 

Electrical  congress  of  1893,  Recommenda- 
tions of,  181 

"          standards  of  measure,  180 
Electricity,  induced,  Quantity  of,  189,  203 
Electrification  by  influence,  78 

phenomena  of,  76 

Electrified  conductors,  Energy  of,  100 
Electrodynamometer,  152 
Electrodynamometer,      Measurement      of 
mean    and    effective    current    by    an, 
200 

Electrolysis,  Application  of  the  conserva- 
tion of  energy  to,  129 
Electromagnet  (defined),  162 

44  cores  traversed  by  variable 

currents,  207 

Electromagnetic    displacements,   Explana- 
tion based  on  the  properties  of  lines  of 
force,  161 
Electromagnetic  induction,  Flux  of  force 

producing,  188 
Electromagnetic    induction,  General  law 

of,  184 

Electromagnetic  induction,  Graphic  repre- 
sentation of  cases  in,  199 
Electromagnetic  induction,  Seat  of  electro- 
motive force  in,  187 
Electromagneti    rotation,  154 
Electromagnets,  Energy  expended  in,  163 
44  Forms  and  construction 

of,  165 

Electrometer,  Absolute,  96 
44  Quadrant,  47 

Electromotive  force  and  difference  of  po- 
tential, Distinction  between,  107 
Electromotive  force  constant,  ease  of,  194 

of  contact,  107 
Electromotive  force  of  induction,  Seat  of, 

187 

forces,  Thermal  and  chem- 
ical, 113 
Electrostatic  capacity  of  conductors,  92 

pressure,  87 

Elementary  magnets,  40 
Element  of  current,  Action  of  a  magnetic 

field  on  an,  138 
Element  of  current,  Work  due  to  the  Dis- 


placement of,  under  the  action  o!   a 
pole,  139 
Energy,  Conservation  of  (potentialkinetic), 

6 
44        conservation    of ;    application    to 

electrolysis,  129 

44        electric,  Present  views  of  the  prop- 
agation of,  224 

44        expended  in  electromagnets,  163 
44        of  a  current  in  a  magnetic  field,  142 
44        of  a  magnetic  shell  in  a  field,  45 
44        of  an  electric  current,  Intrinsic,  144 
"        of  electrified  conductors,  100 
44        of  the  electric  current  (general  ex- 
pression), 124 

44        of  the  electric  current  homogene- 
ous conductor,  125 

44        of  the  electric  current  heterogene- 
ous conductor,  126 
44        of  two  currents,  Relative,  143 
44        of  two  magnetic  shells,  Relative, 

46 
Equilibrium  of  a  body  in  a  magnetic  field, 

65 

Equipotential  surfaces,  13 
Equivalent  sine  curves,  74 
Ether,  Modes  of  motion  of,  6 

Ocean  of,  9 

Ewing's  addition  to  Weber's  hypothesis,  64 
Experiments  with  static  electricity,  80 


Faraday's  disc,  191 
"  law,  127 
44  rule  for  energy  of  current  in 

magnetic  field,  145,  186 
Ferranti  effect,  221 
Ferraris'  arrangement  for  obtaining  rota-" 

tion,  210 

Ferromagnetic  bodies,  51 
Field,  Electric,  (defined),  83 

44      magnetic,  Action  of,  on  an  element 

of  current,  138 

"      Magnetic,  due  to  an  infinite  rectilin- 
ear current,  136 
44      magnetic,  Energy  of  electric  current 

in,  142 

44      magnetic,  Measurement  of,  by  quan- 
tity of  electricity  induced,  192 
44      magnetic,  Measurement  of  intensity 

of  electromagnetic  method,  231 
44      magnetic,  Method  based  on  induc- 
tion, 232 

44          Method  by  oscillation,  230 
44          Modifications  of  the  prop- 
erties of  bodies  in  a,  167 


INDEX. 


387 


Field   of  force,  (defined),  n 
"       "      u     of  single  mass,  14 
"       "      "     of  two  acting  masses,  16 
"       "      "     Uniform,  15 
"       "      "  "         action  of,  on  mag- 

net, 37 

"    uniform  magnetic,  Movable  conduc- 
tor in,  190 

Filament,  Magnetic  or  sdfenoidal,  41 
Flux  of  force,  18 

"     "      "      producing  induction,  188 
Force,  Field  of,  n 
"       Flux  of,  18 
"      Line  of,  (defined),  13 
"      Magnetomotive,  164 
"      Tubes  of,  17 
Forms  and  construction  of  electromagnets, 

165 
Formula,  Frolich's,  58 

"         Muller,  von  Waltenhofen,   and 

Kapp,  59 
Foucault  currents,  206 

"  "         Calculation    of    power 

lost  in,  207 
Frolich's  formula,  58 


Galvanometer,  Tangent,  147 

shunt,  149 
"  Thomson,  148 

with    shunt,  Discharge  of 

condenser  into,  197 
General  theory  of  units,  169 
Graphic  representation  of  Ohm's  law,  n6, 

1 20 

"        representations  of  cases  in  electro- 
magnetic induction,  199 
Grothiiss'  hypothesis,  128 
Guard-ring  condenser,  95 

H 

Hall  effect,  168 

Heterogeneous  circuit,  Application  of  Ohm's 

law  to,  119 

Hypothesis,  Ampere's,  on  the  nature  of  mag- 
netism, 141 
"  Grothiiss',  128 

"  Weber's,  39 

"        Ewing's  addition  to, 

64 
Hysteresis,  57,  66-75 

"          Coefficient  of,  71. 

"         Determination  of,  and  molecular 

magnetic  friction,  69 
"          Loss  due  to,  in  magnetization, 


Hysteretic  loop,  67 

"         loss  of  energy,  70,  75 

I 

Impedance,  213  216 
Impedances,  Joint,  216 
Induced  currents,  Rotation  due  to,  209 
'*        electric  currents,  182 
"        electricity,  Quantity  of,  189,  203 
Inducing  and  induced  currents,  Mutual  re- 
pulsion, 212 

Inductance  and  capacitance,  214 
"          and  reactance,  215 
"          (defined),  213 
"          mutual,  Expression  for,  204 
Induction,  electromagnetic,  Flux  of  force 

producing,  188 
"          electromagnetic,    General    law 

of,  184 

"          electromagnetic,  Graphic  repre- 
sentation of  cases  in,  199 
"          in  metallic  masses,  205 
"          Mutual,  of  two  circuits,  201 

"        "    "    fixed  circuits,  202 
"          Seat  of  electromotive  force  of, 

187 
Infinitely  thin  spherical  shell;  no  action  on 

masses  within  it,  26 
"         Action  on  external  masses  of,  27 
Infinite  rectilinear  current,  Magnetic  field 

due  to  an,  136 
"       rectilinear  current.    Potential  due 

to,  146 

Influence,  Electrification  by,  78 
Instantaneous  electric  discharge.  Measure- 
ment of  an,  150 
Insulators  and  conductors,  77 
Intrinsic  energy  of  an  electric  current,  144 

J 

Jar,  Leyden,  99 
Joint  impedances,  216 
Joule's  effect,  127,  196 

K 

/iTand  /m,  Dimensions  of,  175 
Kelvin  effect,  131 
Kirchhoff  s  laws,  121 

"  "     Application  of,  to  derived 

circuits,  122,  196 

L 

Laplace's  law,  137 
Law,  Becquerel's,  127 
"     Coulomb's,  Nature  of  coefficient  K  in, 

104 
"  "  Proof  of,  49 


388 


INDEX. 


Law,  Elementary, governing  theNewtonian 
forces,  10 

11      Faraday's,  127 

"     of  electric  actions,  82 

"  of  electromagnetic  induction,  Gen- 
eral, 184 

"     of  magnetic  attractions,  34 

"     of  successive  contacts,  112 

"      Laplace's,  137 

"      Lenz's,  183 

"     Ohm's,  114 

"  "  Application  of,  to  the  case  of 
a  heterogeneous  circuit,  119 

"  "  Case  of  a  conductor  of  con- 
stant section,  115 

"          "       Graphic  representation  of,  1 16, 

120 

Laws,  Kirchhoff' s,  121 

Application    of,   to    de- 
rived circuits,  122,  196 
"      of  the  electric  currents;  preliminary 

note,  in 

"      of  thermo-electric  action,  132 
Lenz's  law,  183 
Leyden  jar,  99 
Lightning-rods,  91 

Lines  of  force,  Explanation  of  electromag- 
netic  displacements  based  on  proper- 
ties of,  161 
Loop,  Hysteretic,  67 
Loss  of  energy,  Hysteretic,  70,  75,  193 

M 
Magnet,  (defined),  32 

"      Action  of  earth  on  a,  33 

"      Annular,  54 

"      Cylindrical,  (infinite),  55 

"       Disc,  53 

"      Portative  power  of,  56 

*'      Rotation  of  a  brush  discharge  by 

a,  156 
"       Rotation  of  an  electric  current  by 

a.  155 

"      Sphere,  53 

Magnetic  and  diamagnetic  bodies,  51 
"          circuit,  164 
*'         field,  Action  of,  on  an  element  of 

current  on  a,  138 
"  "     due  to  an  infinite  rectilinear 

current,  136 
"  "     Energy  of  electric  current 

in  a,  142 

"  "     Energy  of  shell  in  a,  45 

"  "      Equilibrium  of  a  body  in,  65 

Magnetic  field,  Measurement  of,  by  quan- 
tity of  electricity  induced,  192 


Magnetic  field,  Electromagnetic  Method  of 

measurement  of  intensity,  231 
Magnetic  field,  Measurement  of  intensity 

of  method  based  on  induction,  232 
Magnetic  field,  Measurement  of  intensity 

of  method  by  oscillation,  230 
Magnetic  field,  Modifications  of  the  proper- 
ties of  bodies  in  a,  167 
Magnetic  field  of  force,  Terrestrial,  38 

'     uniform,  Movable  conductor 

in,  190 

Magnetic  induction,  magnetization,  and 
permeability,  Another  way  of  looking 
at,  60 

Magnetic  moment  of  a  magnet,  Determi- 
nation of,  48 

"          or  solenoidal  filament,  41 
Magnetic    permeability;    measurement    of 

magnetometric  method,  236 
Magnetic    permeability;    measurement   of 

methods  based  on  induction,  233-235 
Magnetic    permeability;    measurement    of 

method  by  portative  power,  237 
Magnetic  pole,  Work  due  to  the  displace- 
ment of  an  electric  circuit  under  the 
action  of  a,  140 

Magnetic  pole,  Work  due  to  the  displace- 
ment of  an  element  of  current  under 
the  action  of,  139 
Magnetic  potential,  36 

"  "  due  to  a  circular  elec- 

tric current,  147 

Magnetic  potential  due  to  an  infinite  recti- 
linear electric  current,  146 
Magnetic  potential  due  to  an  electric  cir- 
cuit, 141 

"  quantities,  (definitions  of),  36 
"  resistance,  or  reluctance,  164 
"  shell,  (defined),  43 

"    in  a  field,  Energy  of  a,  45 
"  "    mentioned,  42 

"     Potential  due  to,  43 
Magnetic  shell,  Work  done  on  unit  posi- 
tive magnetic  mass  in  passing  from  one 
side  to  the  other  of  a,  44 
Magnetic  shells,  Relative  energy  of  two,  46 
Magnetization,    magnetic    induction,    and 
permeability,  Another  way  of  looking 
at,  60 
Magnetization  of  a  conductor  by  a  current, 

1 66 

"  Variations  of,  with  the  mag- 

netizing force,  57 
Work  absorbed  in,  193 
Magnetizing  force,  Variation  of  magnetiza- 
tion with,  57. 


INDEX. 


389 


Magnetizing  force,  Work  done  in,  61 
Magnetism,  Effect  of  temperature  on,  63 
"          Nature  of,  Ampere's  hypothe- 
sis, 141 

Magnetometer,  48 
Magnetomotive  force,  164 
Magnets,  Artificial,  47 

Elementary,  40  d, 

Uniform,  42          •*• 

Maxwell  and  Pirani's  method  for  measuring 
self-induction  in  terms  of  a  capacity,  239 
Maxwell  and  Rayleigh's  method  for  meas- 
uring a  coefficient  of  self-induction  in 
terms  of  a  resistance,  238 
Maxwell's  rule  for  energy  of  current  in  a 

magnetic  field,  142 

Maxwell's  rule  for  relation   between    the 
directions  of  current  and   rotation  of 
magnetic  north  pole,  135,  185 
Mean  current  and  effective  current,  Meas- 
urement of,  by  dynamometer,  200 
Measurement  of  a  coefficient  of  self-induc- 
tion,  Ayrton   and  Perry's  method  for 
comparing  with  a  standard  coil,  240 
Measurement  of  a  coefficient  of  self-induc- 
tion in  terms  of  a  resistance,  Maxwell 
and  Rayleigh's  method,  238 
Measurement  of  a  coefficient  of  self-induc- 
tion, Maxwell's  method,  in  terms  of  a 
capacity,  239 
Measurement  of  angles  in  radians,  50 

of  an  instantaneous  electric 

discharge,  150 

of    power,   continuous   cur- 
rent, 225 
Measurement  of  magnetic  field   intensity; 

electromagnetic  method,  231 
Measurement  of  magnetic  field  intensity; 

method  based  on  induction,  232 
Measurement  of  magnetic  field  intensity; 

method  by  oscillation,  230 
Measurement   of    magnetic    permeability; 

magnetometric  method,  236 
Measurement   of   magnetic    permeability; 

methods  based  on  induction,  233-225 
Measurement    of   magnetic    permeability; 

method  by  portative  power,  237 
Measurement  of  mutual  induction,  Carey- 
Foster  method,  241 
Mechanical  action,  Effect  of,  on  hysteresis, 

71,  105 

Metallic  masses,  Induction  in,  205 
Method,  Alternating  current  for  determin- 
ing hysteresis,  69 

"       Ballistic  current  for  determining 
hysteresis,  69. 


Method,  Ayrton  and  Perry's,  for  compar- 
ing a  coil's  self-induction  with  that  of 
a  standard  coil,  240 

Method,  Carey-Foster,  for  mutual   induc- 
tion, 241 

Electromagnetic,   for   measuring 

magnetic  field,  231 
Induction,    for   measuring    mag- 
netic field,  232 

Oscillation,  for    measuring  mag- 
netic field,  230 

Methods  for  measuring  magnetic  permea- 
bility; induction,  233-235 
Methods  for  measuring  magnetic  permea- 
bility; magnetometric,  236 
Methods  for  measuring  magnetic  permea- 
bility; portative  power,  237 
Methods     for     measuring     self-induction, 
Maxwell  and  Rayleigh's,  in  terms  of  a 
resistance,  238 

Methods  for  measuring  self-induction, 
Maxwell  and  Pirani's,  in  terms  of  a  ca- 
pacity, 239 

Molecular  magnetic  friction   Effect  of,  73 
(defined),  68 
determination  of,  69 

Moment,  magnetic,  Determination  of,  48 
Motor,  Ferraris',  210 

Movable  conductor  in  a  uniform  field,  190 
Mutual  action  of  electric  currents,  159 
"     inductance,  Expression  for,  204 
"     induction,  Carey-Foster  method,  241 
of  two  circuits,  201 
"    "    fixed  circuits,  202 

N 

Nature  of  magnetism;  Ampere's  hypothe- 
sis, 141 
Nomenclature  of  practical  units,  178 

Proposed,  for  units,  174 
Number  of  lines  of  force,  24 
Numerical  relation  of  B  and  /*,  62 

"          value  and  magnitude  of  units, 
reciprocal  relation  of,  2 


Objective  method,  Sir  Wm.  Thomson's,  50 
Oersted's  discovery,  135 
Ohm,  International,  180 

"      Legal,  1 80 
Ohm's  law,  114 
"         "     Application    to  the  case  of  a 

heterogeneous  circuit,  119 
"         "     case  of  conductor  of  constant 
section,  115 


390 


INDEX. 


Ohm's  law,  Graphic  representation  of,  116, 

120 

Oscillating  discharge,  222 


Peltier  effect,  126,  130 

Periodic  current,  Power  of,  227 

"  "        "   inductive    con- 

ductors, 229 

Periodic  current,  Power  of,  non-inductive 
conductors,  228 

Period  of  the  current,  Variable,  117 

Period  of  the  current,  Variable;  applica- 
tion of  Ohm's  law  in  but  slightly  con- 
ductive bodies,  118 

Permeability,  magnetic,  Measurement  of; 
magnetometric  method,  236 

Permeability,  magnetic,  Measurement  of; 
methods  based  on  induction,  233-235 

Permeability,  magnetic,  Measurement  of; 
method  by  portative  power,  237 

Permeability,  magnetization  and  magnetic 
induction,  Another  way  of  looking  at,6o 

Pile,  Thermo-electric,  134 

Pirani's  modification  of  Maxwell's  method 
of  measuring  a  self-induction  in  terms 
of  a  capacity,  239 

Phenomena  of  electrification,  76 

Phenomena  which  accompany  the  propaga- 
tion of  current  in  a  conductor,  217 

Plate  condenser,  94 

Points,  electrified,  Discharging  power  of,  89 

Pole,  magnetic,  Work  due  to  the  displace- 
ment of  an  electric  circuit  under  the 
action  of,  140 

Pole,  magnetic,  Work  due  to  the  displace- 
ment of  an  element  of  current  under 
the  action  of,  139 

Pole,  Unit,  magnetic,  (defined),  35 

Portative  power  of  a  magnet,  56 

Potential  (defined),  12 

44        constant   at  all  points  within  a 

spherical  shell,  26 
•*        electric,  (defined),  83 
44         of  conductor  in  equilibrium,  84 
44         Electric,  of  the  earth,  85 
**         energy  of  masses  subjected    to 

Newtonian  forces,  25 
**        in  a  conductor.  Means  for  keeping 
up  a  constant  difference  of,  113 
*'        due  to  infinitely    thin  disc  uni- 
formly charged,  31 

"         Magnetic,  due  to  a  circular  elec- 
tric current,  147 

*'         Magnetic,  due  to  an  electric  cir- 
cuit, 141 


Potential  Magnetic,  due  to  an  infinite  recti- 
linear current,  146 

Power  and  torque  tests  for  hysteresis,  69 
"       of  periodic  current,  227 

"         inductive     con- 

ductors,  229 
44        "    periodic  current,  non-inductive 

conductors,  228 

14        *l      continuous    current,   Measure- 
ment, 225 
44        "     points  on  electrified  conductor, 

89 

44       Portative,  of  a  magnet,  56 
Powers,  Thermo-electric,  133 
Practical  and  C.  G.  S.  systems,  Relation  be- 
tween, 177 

*'         system  of  units,  177 
"         units,  Nomenclature  of,  178 
Present  views  on  the  propagation  of  elec- 
tric energy,  224 
Pressure,  Electrostatic,  87 
Proof  of  Coulomb's  law,  49 
Propagation  of    current  in   a    conductor, 

Phenomena  which  accompany,  217 
Propagation   of   electric    energy,  Present 

views  of,  224 

Properties  of  bodies  in  a  magnetic  field, 
Modifications  of,  167 

Q 
Quadrant  electrometer,  47 

charged    at     high 

potential,  102 

"  "  Theory  of,  101 

Quantity  of  induced  electricity,  189,  203 
Quantity  of  electricity  induced,   Measure- 
ment of  intensity  of  magnetic  field  by, 
192 

R 

Radians,  50 

Ratio  "  z/,"  Value  of,  176 

"  Rational  "  system  of  units,  179 

Rayleigh  and  Maxwell's  method  for  meas- 
uring self-induction  in  terms  of  a  re- 
sistance, 238 

Reactance  and  inductance,  215 

Reactions  produced  in  a  circuit  traversed 
by  a  current,  160 

Recalescence,  63 

Rectilinear  current,  infinite,  Magnetic  field 

due  to,  136 

44        infinite,   Potential  due 
to,  146 

Relation  between  practical  and  C.  G.  S.  sys- 
tems, 177 


INDEX. 


39* 


Relation  between  the  directions  of  current 
and  rotation  of  magnetic  north  pole, 

i35.  l85 
Relative  energy  of  two  currents,  143 

"  "       "      "    magnetic  shells,  46 

Reluctance  or  magnetic  resistance^  164 
Repulsion  exercised  by  an  inducing  current 

on  an  induced  current,  212 
Residual  discharge  of  a  condenser,  106 
Resistance,  Magnetic,  or  reluctance,  164 
Reversing  a  current,   Rotation  produced 

by,  158 

Ring-shaped  bobbin,  153 
Rotation,  Electromagnetic,  154 

'*        of  a  brush  discharge  by  a  magnet, 
156 

"         of  a  current  by  a  magnet,  155 

"         produced  by  reversing  a  current, 

iS3 

"         under  the  action  of  induced  cur- 
rents, 209 
Rule,  Faraday's,  for  energy  of  current  in 

magnetic  field,  145,  186 
"      Maxwell's,  for  energy  of  a  current, 

in  a  magnetic  field,  142 
Rule,  Maxwell's,  for  relation  between  the 
directions  of  current  and  rotation  of 
magnetic  north  pole,  135,  185 


Screen,  Electric,  90 

Seat  of  electromotive  force  of  induction,  187 

Seebeck  effect,  130 

Self-induction  and  capacity,  Combined 
effects  of,  on  a  circuit  traversed  by  al- 
ternating currents,  221 

Self-induction  and  capacity  of  a  circuit, 
Comparative  effects  of  the,  219 

Self-induction,  Ayrton  and  Perry's  method 
for  comparing  with  a  standard  coil,  240 

Self-induction  in  a  circuit  composed  of  linear 

conductors,  194 
in  terms  of  a  capacity,  Max- 
well's method  for  measuring,  230 

Self-induction  in  terms  of  a  resistance,  Max- 
well and  Rayleigh's  method  for  measur- 
ing a  coefficient  of,  238 

Self-induction  in  the  mass  of  a  cylindrical 
conductor,  208 

Self-induction  of  a  circuit  composed  of 
linear  conductors,  where  there  is  a  peri- 
odic or  undulatory  electromotive  force, 
198 

Self-induction  of  a  circuit,  163 

Shallenburger's  meter,  211 

Shell  magnetic,  Energy  of,  in  field,  45 


Shell  magnetic,  Work  done  on  unit  positive 
magnetic  mass  in  passing  from  one  side 
to  the  other  of,  44 

Shells  magnetic,  Relative  energy  of  two,  46 
"  "          (defined),  43 

"  "          mentioned,  42 

"  "          potential  due  to,  43 

Shunt,  Galvanometer,  149 

Siemens  unit,  180 

"       wattmeter,  226 

Sine  curves,  Equivalent,  74 

Slightly  conductive  bodies,  Application  of 
Ohm's  law  to  the  variable  period  of 
current  in,  118 

Solenoid,  Cylindrical,  151 

Solenoidal  or  magnetic  filament,  41 

Spark,  Electric,  no 

Special  characteristics  shown  byalternating 
currents  (mode  of  combining),  218 

Specific  inductive  capacity  of  dielectrics, 
103 

Sphere,  homogeneous,  Action  of,  upon  ex- 
ternal point,  28 

Sphere,  homogeneous,  Action  of,  upon  in- 
ternal point,  29 

Sphere,  homogeneous,  Surface  pressure  on, 
3° 

Spherical  condenser,  93 
magnet,  53 

Static  electricity,  Experiments  with,  80 

Standards  of  measure,  Electrical,  180 

Successive  contacts,  Law  of,  112 

Surface  density  of  magnetism,  40 

"      distribution  of  electric  charge  in 

equilibrium,  81 

"      pressure  on  homogeneous  sphere, 
3° 

System  of  units,  C.  G.  S.,  171 
Practical,  177 
"Rational,"  179 

Systems  of  units  in  terms  of  K  and  /x,  173 


Table  of  coefficients  of  hysteresis,  71 
"      "  values  of  B  and  /x,  62 
"     "       "       "  coefficients  of  hysteresis, 

62 

Tangent  galvanometer,  147 
Temperature,  Effect  of,  on  magnetism,  63 
Terrestrial  magnetic  field  of  force,  38 
Theorem,  19 

"         Coulomb's,  86 
"        Gauss',  20 

"        corollary  I.,  21 

11,22 

III,  a* 


392 


INDEX. 


Theory  of  units,  169 

"        "  quadrant  electrometer,  101 
Thermal  electromotive  forces,  113 
Thermo-electric  action,  Laws  of,  132 

pile,  134 

"  powers,  133 

Thomson  galvanometers,  148 

"         Sir  Wm.,  Objective  method,  50 
Time-const-nt,  194 

Transmission  of  electric  waves  in  the  sur- 
rounding medium,  223 
Tube  of  electric  force,  Corresponding  ele- 
ments of,  88 
"      "  force,  17 
"      "      "      unit,  24 

U 

Uniform  magnetic  field,  Movable  conductor 

in,  190 

"        magnets,  42 
Unit,  British  Association,  180 
"    Siemens,  180 
"    magnetic  pole,  (defined),  35 
Units,  C.  G.  S.  system  of,  171 
"      derived,  2 

"         example  of,  3 
"         dimensions  of,  4 
"         mechanical,  5 
"      multiples  and  submultiples  of,  7 
"      C.  G.  S.  system  defined,  2 
"      Dimensions  of,  170 
"      General    considerations    regarding, 
172 


Units,  General  theory,  169 

practical,  Nomenclature  of,  178 
proposed,  Nomenclature  of,  174 
Practical  system  of,  177 
"  Rational  "  system  of,  179 
"      Systems  of,  in  terms  of  JiT  and  p,  173 


Value  of  the  ratio  "  z/,"  176 
Variable  period  of  the  current,  117 
Voltaic  cell,  129 

W 

Wattmeter,  Siemens,  226 

Waves,   electric,   Transmission  of,   in  the 

surrounding  medium,  223 
Weber's  hypothesis,  39 

Ewing's  addition  to,  64 
Wheatstone's  bridge,  123 
Wheel,  Barlow^,  157 
Work  absorbed  in  magnetization,  193 

"     accomplished  on  closing  a  circuit,  195 
Work  done  on  unit  positive  magnetic  mass 

in  passing  from  one  side  Jof  magnetic 

shell  to  the  other  side,  44 
Work  due  to  the  displacement  of  a  circuit 

under  the  action  of  a  pole,  140 
Work  due  to  the  displacement  of  an  element 

of  current  under  the  action  of  a  pole, 

139 
Work  spent  in  magnetizing,  61 


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...  OF  ... 

ENGINEERING  BOOKS 

..  o  ON  .  .  . 

POWER    TRANSMISSION 

ELECTRIC    LIGHTING 

ELECTRIC    RAILWAYS 

THE  TELEPHONE 

THE  TELEGRAPH 

ELECTROCHEMISTRY 

DYNAMOS     AND    MOTORS 

ELECTRICAL    MEASUREMENTS 

TRANSFORMERS 
STORAGE    BATTERIES 

WATER    WORKS 
BRIDGE    ENGINEERING 

COMPRESSED    AIR 
STRUCTURAL    ENGINEERING 

STEAM    POWER    PLANTS 

VENTILATION    AND    HEATING 

STEAM    AND    HOT    WATER    HEATING 

PLUMBING 

METALLURGY 

MARINE    ENGINEERING 


McGRAW  PUBLISHING  COMPANY. 


BOOKS  AND   PAMPHLETS. 

Abbott,  A.  V.,  Telephony.    6  Vols. 


Vol.  i.  Location  of  Central  Offices 

Vol.  2.  Construction  of  Underground  Conduits. 

Vol.  3.  The  Cable  Plant 

Vol.  4.  Construction  of  Aerial  Lines 

Vol.  5.  The  Substation. 


50 
50 
50 
50 
50 

Vol.  6.     Switchboards  and  the  Central  Office I  50 

Price  of  the  set  of  6  vols.  JSG.OO  ;  single  vols.  S1.5O  each. 

American  Plumbing  Practice 

Descriptive  articles  upon  current  practice  with  questions  and 
answers  regarding  the  problems  involved.  268  pages,  536 
illustrations 2  50 

American  Steam  and  Hot-Water  Heating  Practice 

Descriptive  articles  upon  current  practice  with  questions  and 
answers  regarding  the  problems  involved.  268  pages.  536 
illustrations 3  oo 

American  Street  Railway  Investments 

Financial  data  of  over  1,300  American  city,  suburban  and  inter- 
urban  electric  railways,  statistics  of  operation,  details  of  plant 
and  of  equipment,  and  names  of  officers.  Published  annually. 

Vol.  X,  1904  edition $5  oo 

We  can  still  supply  Vols.  I  (1894)  to  X  (1903)  at  $5.00  each. 

Badt,  F.  B.— Incandescent  Wiring  Handbook 

With  42  illustrations  and  5  tables.     Fifth  edition i  oo 

New  Dynamo  Tenders'  Handbook 

With  140  illustrations,  226  pages i  oo 

Bell  Hangers'  Handbook 

With  98  illustrations.     Third  edition.     105  pages i  oo 

Baldwin,  William  J.-Hot-Water  Heating  and  Fitting 

385  pages,  200  illustrations 2  50 

Baum,  F.  G.— Alternating  Current  Transformer 

195  pages  and  122  illustrations i  50 

Bedell,  Fred'k,  and  A.  C.  Crehore— Alternating  Currents 

An  analytical  and  graphical  treatment  for  students  and  engi- 
neers. Fourth  edition,  cloth,  325  pages,  no. illustrations  and 
diagrams 2  50 


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^Behrend,  B.  A.— The  Induction  Motor 

105  pages,  56  illustrations ,...«...$!  5° 

Bell,  Louis— The  Art  of  illumination 

350  pages,  125  illustrations'. 2  50 

Electric  Power  Transmission 

The  best  treatise  yet  written  on  the  practical  and  commercial 
side  of  electrical  power  transmission.  Third  edition.  632 
pages,  285  illustrations,  21  plates 3  °° 

Power  Distribution  for  Electric  Railways 

Deals  with  the  laying  out  and  calculating  of  electric  railway 
circuits.  Substation  distribution  is  also  considered.  Third 
edition.  303  pages,  148  illustrations 2  50 

Billings,  W.  R.— Some  Details  of  Water-Works  Construc- 
tion 

96  pages,  28  illustrations. . . . , 2  oo 

Billings,  John  S.— Ventilation  and  Heating 

500  pages,  210  illustrations 4  oo 

Bjorling,  Philip  R.— Briquettes  and  Patent  Fuel 

Their  manufacture  and  machinery  connected  therewith.  254 
pages.  121  illustrations 3  75 

Camp,  W,  M.— Notes  on  Track  Construction  and  Mainte- 
nance 

New  edition.     Extended  and  enlarged.     More  than  1200  pages 

and  600  illustrations.     In  two  vols.  $4.00;  or  bound  in  one  vol..  3  75 

Cement  Industry,  The 

Description  of  Portland  and  natural  cement  plants  in  the 
United  States  and  Europe,  with  notes  on  materials  and  pro- 
cesses in  Portland  cement  manufacture.  235  pages,  132  il- 
lustrations    3  oo 

Collins,  A.  F.— Wireless  Telegraphy    (in  preparation.) 
Considere,  A.— Reinforced  Concrete 

Translated  from  the  French,  with  a  preface  and  additions  by 
Leon  S.  Moisseiff 2  oo 

Crehore,  A.  C.-(See  Bedell,  Fred'k) 

Cushing,  H.  C.— Standard  Wiring  for  Electric  Light  and 
Power,  1904  Edition 

As  adopted  by  the  Fire  Underwriters  of  the  United  States. 
Flexible  leather,  144  pages,  illustrated  . . . I  oo 


McGRAW  PUBLISHING  COMPANY. 


de  la  Tour,   Boy— The  Induction   Motor;    Its  Theory  and 
Design 

Set  forth  by  a  practical  method  of  calculation.  Translated 
from  the  French  by  C.  O.  Mailloux.  225  pages  and  77  illus. .  .$2  50 

Diagrams    of    Electrical    Connections,    with    Explanatory 
Text 

32  pages,  paper 50 

Dobbs,  A.  EB— Practical  Features  of  Telephone  Work 

I2mo,  cloth.     134  pages,  61  illustrations 75 

Electrical  Designs 

262  pages,  289  illustrations 2  oo 

Emmet,  William  LeRoy— Alternating  Current  Wiring  and 
Distribution 

Contains  a  very  clear  account  of  the  principles  of  alternating 
currents  from  the  practical  point  of  view,  and  of  their  distri- 
bution and  application  to  lighting  and  power.  Second  edition. 
98  pages,  33  illustrations.  i6mo,  cloth  I  oo 

Fairman,  James  F.— Telephone  Wiring    (in  preparation.) 

Freund,  Leopold— Elements  of  General  Radio-Therapy  for 
Practitioners 

Translated  by  G.  H.  Lancashire.  538  pages,  106  illustrations, 
with  an  appendix  of  59  pages  and  86  illustrations  on  "Notes 
on  Instrumentation,"  by  Clarence  A.  Wright 5  oo 

Gerard  Eric— Electricity  and  Magnetism 

With  chapters  by  C.  P.  Steinmetz,  A.  E.  Kennelly  and  Dr. 
Cary  T.  Hutchinson.  Translated  under  the  direction  of 
Dr.  Louis  Duncan.  392  pages,  112  illustrations  2  50 

Gonzenbach,  E.— Engineering  Preliminaries  for  an  Inter- 
urban  Electric  Railway 

72  pages,  illustrated    , I  oo 

Goodell,  John— Water-Works  for  Small  Cities  and  Towns 

281  pages,  53  illustrations 2  oo 

Gotshall,  W.  C.— Notes   on    Electric    Railway    Economics 
and  Preliminary  Engineering 

A  series  of  Notes  on  the  Preliminary  Engineering  of  Inter- 
urban  Electric  Railways.  260  pages,  illustrated 2  oo 


McGRAW  PUBLISHING  COMPANY. 


Hanchett,  George  T.— Modern  Electric  Railway  Motors 

The  various  commercial  types  of  electric  railway  motors  are 
described  in  full  detail,  including  mounting  and  control.  200 
pages,  157  illustrations  .  i $2  oo 

Handy  Tables  for  Electrical  and  Steam  Engineers 

18  tables  found  to  be  useful  in  arriving  at  approximate  results 
without  extensive  calculation.  Pamphlet,  8vo,  16  pages 50 

Hendricks,   S.   E.— Commercial    Register   of  the    United 
States.     I  3th  Annual  Edition,  I  904 

Especially  devoted  to  the  interests  of  the  architectural,  mechani- 
cal, engineering,  contracting,  electrical,  railroad,  iron,  steel, 
mining,  mill,  quarrying  and  kindred  interests.  1228  pages. . . .  6  oo 

Hering,  Carl— Electrochemical  Equivalents i  oo 

Ready  Reference  Tables.    Vol.   1 

Conversion  factors  of  every  unit  of  measure  in  use,  based  on  the 
accurate  legal  standard  values  of  the  United  States.  Flexible 
leather,  pocket  size.  196  pages 2  50 

The  Universal  Wiring  Computer 

For  determining  the  sizes  of  wires  for  incandescent  electric 
lamp  leads,  and  for  distribution  in  general  without  calculation, 
with  some  notes  on  wiring  and  a  set  of  Auxiliary  Tables.  44 
pages,  4  charts , I  oo 

Herrick,  Albert  B.— Practical  Electric  Railway  Hand  Book 

Handsomely  bound  in  leather,  with  flap.     450  pages 3  oo 

Hopkinson,  John— Original  Papers  on  Dynamo  Machinery 
and  Allied  Subjects 

Authorized  American  edition.     249  pages,  98  illustrations I  oo 

Houston,  E.  J.— Advanced  Primers  of  Electricity 

Vol.  i.  Electricity  and  Magnetism  and  other  Advanced 

Primers.  Second  edition,  318  pages i  oo 

Vol.  2.  Electrical  Measurements  and  other  Advanced  Primers. 

429  pages I  oo 

Vol.  3.  Electrical  Transmission  of  Intelligence  and  other  Ad- 
vanced Primers.  330  pages i  oo 

Dictionary  of  Electrical  Words,  Terms  and  Phrases 

Fourth  edition.     Greatly  enlarged.     990  double  column  octavo 

pages,  582  illustrations 7  oo 

Pocket  edition.     945  pages,  cloth 2  50 

Pocket  edition.     945  pages,  leather 3  oo 


McGRAW  PUBLISHING  COMPANY. 


Electricity  One  Hundred  Years  Ago  and  To-day 

199  pages,  illustrated $i  oo 


Houston,  E.  J.,  and  A.  E.  Kennelly— Algebra  Made  Easy 

101  pages,  5  illustrations 75 

—    Electricity   Made   Easy   by    Simple   Language   and 
Copious  Illustrations 

348  pages,  207  illustrations I  50 

The  Interpretation  of  Mathematical  Formulae 

225  pages,  9  illustrations i  25 

Electrical     Engineering    Leaflets,    Containing   the 

Underlying  Principles  of  Electricity  and  Magnetism 

ELEMENTARY  GRADE— suited  to  the  study  of  Electrical 
Artisans,  Wiremen  and  to  Elementary  Students.  296  pages, 

121  illustrations,  8vo,  cloth I  50 

INTERMEDIATE    GRADE suited    to   students    in    High 

Schools  or  Colleges,  and  those  beginning  the  study  of  Electrical 

Engineering.     300  pages,  140  illustrations,  8vo,  cloth I   50 

ADVANCED  GRADE— suited  to  students  taking  Technical 
Courses  in  Electrical  Engineering.  296  pages,  121  illustrations, 
8vo,  cloth , i  50 

Electro-Dynamic  Machinery 

A  text-book  on  continuous-current  dynamo-electric  machinery 
for  electrical  engineering  students  of  all  grades.  331  pages, 
232  illustrations 2  50 

Recent  Types  of  Dynamo-Electric  Machinery 

612  pages,  435  illustrations 4  oo 

(ELECTROTECHNICAL    SERIES     1O    VOLUMES) 

Alternating  Electric  Currents 

Third  edition,  271  pages,  102  illustrations   i  oo 

Electric  Arc  Lighting 

Second  edition.     437  pages,  172  illustrations i  oo 

Electric  Heating 

290  pages,  86  illustrations  i  oo 

Electric  Incandescent  Lighting 

Second  edition.     508  pages,  161  illustrations i  oo 

The    Electric    Motor 

377  pages,  122  illustrations i  oo 


McGRAW  PUBLISHING  COMPANY. 


Electric  Street  Railways 

367  pages,  158  illustrations  .  . , $i  oo 

Electric  Telegraphy 

448  pages,  163  illustrations   i  oo 

The  Electric  Telephone 

Second  edition.     454  pages,  151  illustrations I  oo 

Electro-Therapeutics 

Second  edition.     452  pages,  147  illustrations i  oo 

Magnetism 

294  pages,  94  illustrations i  oo 

International  Electrical  Congress:   Transactions  and  Pro- 
ceedings 

(a}     At  the  World's  Fair,  Chicago,  1893.     One  volume 3  oo 

(b)  At  the  St.  Louis  Exposition,  1904.  To  be  published  after 
the  Congress,  probably  about  December,  1904,  in  three  paper 
bound  volumes 10  oo 

Kennelly,  A.  E. — (See  Houston,  E.  J.  and  A.  E.  Kennelly.) 
Lancashire,   G.   H.— (See  Freund,  Leopold) 

LeChatelier,  H.— Experimental  Researches  Upon  the  Con- 
stitution of  Hydraulic  Mortars 

Translated  from  the  French  by  J.  L.  Mack,  with  an  appendix...   2  oo 

Lyndon,  Lamar— Storage  Battery  Engineering 

A  practical  treatise  for  Engineers.  Second  edition,  cloth,  360 
pages,  178  illustrations  and  diagrams,  4  large  folding  plates. . . .  3  oo 

Mack,  J.  L.-(SeeH.  LeChatelier) 
MailloUX,   C.  O.— (See  de  la  Tour.) 

Mason,  H.— Static  Electricity 

Cloth,  155  pages,  63  illustrations 2  oo 

Merrill,  A.  E.— Electric  Lighting  Specifications 

For  the  use  of  Engineers  and  Architects.  Second  edition,  en- 
tirely rewritten.  213  pages I  50 

Reference  Book  of  Tables  and  Formulas  for  Elec- 
tric Railway  Engineers 

Second  edition.  Flexible  morocco.  128  pages.  Interleaved 
with  blank  pages  I  oo 


McGRAW  PUBLISHING  COMPANY. 


Meyer,  Henry  C.,  Jr.— Steam  Power  Plants,  their  Design 
and  Construction 

160  pages,  16  plates  and  65  illustrations $2  oo 

Meyer,  Henry  C.— Water  Waste  Prevention:   Its  Import- 
ance, and  the  Evils  Due  to  Its  Neglect 

Cloth.     Large,  8vo I  oo 

Miller,  Kempster  B.— American  Telephone  Practice 

A  comprehensive  treatise,  including  descriptions  of  apparatus, 
line  construction,  exchange  operation,  etc.  Third  edition.  518 

pages,  379  illustrations 3  oo 

NOTE. — The  FOURTH  EDITION,  greatly  enlarged  and  almost 
entirely  rewritten,  about  750  pages,  and  500  new  illustrations, 
will  be  ready  in  October  or  November,  1904.  The  price  is  not 
yet  fixed. 

Moisseiff,  L.  8.— (See  Considere.) 

Monroe,  William  S.— Steam  Heating  and  Ventilation 

150  pages,  90  illustrations 2  oo 

More,  James,  and  Alex.  M.  McCallum— English  Methods 
of  Street  Railway  Track  Construction 

Reprinted  from  the  Street  Railway  Journal.  Pamphlet.  26 
pages,  illustrated 35 

Ohly,  J.— Analysis,   Detection  and   Commercial   Value  of 
Rare  Metals 

A  treatise  on  the  occurrence  and  distribution  of  the  Rare  Metals 
and  Earths,  the  method  of  determination  and  their  commer- 
cial value  in  the  arts  and  industries,  with  a  historical  and  statis- 
tical review  of  each.  216  pages.  First  edition,  with  a  supple- 
mentary chapter 3  O° 

Parham,  E.  C.,  and  J.  S.  Shedd— Shop  and   Road  Testing 
of  Dynamos  and  Motors 

626  pages,  21 1  illustrations 2  50 

^Poincare,  H.— Maxwell's  Theory  and  Wireless  Telegraphy 

Part  I. — Maxwell's  Theory  and  Hertzian  Oscillations,  by  H. 
Poincare,  translated  by  Fred'k  K.  Vreeland.  Part  II.— The 
Principles  of  Wireless  Telegraphy,  by  Fred'k  K.  Vreeland. 
Cloth,  260  pages,  145  illustrations 2  oo 

Poole,  C.  P.— Wireman's  Pocket  Book 

In  press;  ready  shortly i  oo 

Portraits  of  Founders  of  Electrical  Science 

From  Gilbert  to  Henry.  Includes  portraits  of  Gilbert,  Guericke, 
Franklin,  Galvani,  Volta,  Davy,  Ampere,  Faraday,  Henry  and 
Becquerel 29 


McGRAW  PUBLISHING  COMPANY. 


Pratt,  Mason  D.,and  C.  A.  Alden— Street  Railway  Roadbed 

A  treatise  on  the  construction  of  the  roadbed,  giving  data  as 
to  rails,  method  of  track  fastening  and  making  joints,  guard 
rails,  curves,  etc.  13^  pages,  157  illustrations $2  oo 

Reed,  Lyman  C.— American  Meter  Practice 

Cloth,  250  pages,  illustrated 2  oo 

Robinson,  F.  J.— Keys  for  the  Practical  Electrical  Worker 

The  Electric  Light,  Power,  Street  Railway,  Telephone  and  the 
Telegraph  explained  and  illustrated  by  drawings  and  diagrams 
of  connections  from  the  latest  practice.  196  pages,  flexible 
leather  cover  2  oo 

Saunders,  W.  L.— Compressed  Air 

A  cyclopedia  containing  practical  papers  on  the  production, 
transmission  and  use  of  compressed  air.  1188  pages,  498  illus.  5  oo 

Shepardson,  George  D.— Electrical  Catechism 

450  pages,  325  illustrations 2  oo 

Skinner,  Frank  W.— Types   and    Details   of   Bridge   Con- 
struction 

Vol.  I. — ARCH  BRIDGES:  comprising  (i)  Wood  and  Iron  Arch 
Spans;  (ii)  Spandrel  Traced  Arched  Spans;  (iii)  Arch  Truss 
Spans,  and  (iv)  Plate  Girder  Arch  Spans.  Cloth,  306  pages, 
numerous  illustrations , . .  3  oo 

Smith,  Chas.  F.— The  Practical  Testing  of  Dynamos  and 
Motors 

231  pages,  83  illustrations 2  oo 

Stein metz,   Charles   Proteus  — Theoretical   Elements  of 
Electrical  Engineering 

Second  edition.     320  pages,  148  illustrations 2  50 

The  Theory  and  Calculation  of  Alternating  Cur- 
rent Phenomena 

Third  edition.     525  pages,  210  illustrations 4  oo 

Vreeland,  Frederick  K.— (See  Poincare,  H.) 

Wait,  John  C.— Calendar  of  Invention  and  Discovery 

Being  a  memorial  of  the  personal  of  the  greatest  inventors,  dis- 
coverers and  scientists  who  have  contributed  to  the  industrial 
progress  of  the  world.  Cloth,  with  flap i  oo 

Webb,  Herbert  Laws— The  Telephone  Handbook 

New  edition.     160  pages,  133  illustrations I  oo 


io  McGRAW  PUBLISHING  COMPANY. 


Wiener,  A.  E.— Practical   Calculation   of  Dynamo-Electric 
Machines 

A  manual  for  Electrical  and  Mechanical  Engineers,  and  a  Text- 
book for  students  of  Electrotechnics.  Second  edition.  727 
pages,  381  illustrations $3  oo 

Williams,   Chisholm  — High    Frequency   Currents    in    the 
Treatment  of  Disease 

222  pages,  74  illustrations 2  75 

Wright,  Clarence  A.— (See  Freund,  Leopold) 


PERIODICALS. 

Electrical  World  and  Engineer 

An  illustrated  Weekly   Review  of  Current  Progress  in   Elec- 
tricity and  its  Practical  Applications.     Annual  subscription.. . .   3  oo 
To  Foreign  Countries 6  oo 

General  Index  to  the  Electrical  World    (Subject  and  Author.) 

Vol.  I.      From  January  i,  1883,  to  January  I,  1897.    372  pages. .   8  oo 
Vol.  II.    From  January  i,  1897,  to  January  i,  1903.      (In  prepa- 
ration.) 

American  Electrician 

Monthly.     Annual  subscription i  oo 

To  Foreign  Countries 2  oo 

Street  Railway  Journal 

Weekly.     Annual  subscription 3  oo 

To  Foreign  Countries •. 6  oo 

Electric  Railway  Directory  and  Buyers'  Manual 

Published  three  times  a  year.  Annual  subscription  (Street 
Railway  Journal  subscribers  only) I  oo 

General  Index  to  the  Street  Railway  Journal 

By  Subjects  and  Authors.  From  October,  1884,  to  December, 
1903,  including  Vols.  I  to  XXII.  Cloth,  166  large  double  col- 
umn pages 5  oo 

The  Engineering  Record 

Weekly.     Annual  subscription 3  oo 

To  Foreign  Countries 6  oo 

Central  Station  List  and  Manual  of  Electric  Lighting 

Quarterly.     Annual  subscription 4  oo 

Temporary  Binders 

For  any  of  the  above  periodicals I  oo 


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Engineering 
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